Thin-Walled Structures 63 (2013) 13–26

Contents lists available at SciVerse ScienceDirect

Thin-Walled Structures

0263-82

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journal homepage: www.elsevier.com/locate/tws

EBF3PanelOpt: An optimization framework for curvilinearblade-stiffened panels

Sameer B. Mulani n, Wesley C.H. Slemp 1, Rakesh K. Kapania

Department of Aerospace and Ocean Engineering, Virginia Tech, 215 Randolph Hall (0203), Blacksburg, VA 24061, USA

a r t i c l e i n f o

Article history:

Received 26 October 2011

Received in revised form

12 September 2012

Accepted 25 September 2012Available online 23 November 2012

Keywords:

Curvilinear blade stiffeners

Uniform and non-uniform panel thickness

Multiple load-cases

Flat rectangular and cylindrical panel

Buckling

Crippling and von Mises stress

Particle swarm optimization

31/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.tws.2012.09.008

esponding author. Tel.: þ1 540 231 7892; fax

ail addresses: smulani@vt.edu (S.B. Mulani),

@vt.edu (W.C.H. Slemp), rkapania@vt.edu (R.

: http://www.aoe.vt.edu/people/faculty/rkapa

rrently at Orbital Sciences Corporation.

a b s t r a c t

A new framework, EBF3PanelOpt, is being developed for design and optimization of complex, multi-

functional, aircraft structural concepts like pressurized non-circular fuselage structures to be used in

hybrid wing/body vehicles that are subjected to complex structural loading cases. This tool can be used

to integrate materials and structural concepts to exploit emerging additive manufacturing processes

that offer the ability to efficiently fabricate complex structural configurations. Commercial software

packages, MD-Patran (geometry modeling and mesh generation), MD-Nastran (Finite Element Analy-

sis), are integrated in the EBF3PanelOpt framework using Python programming environment to design

stiffened panels with curvilinear stiffeners. Typically, these panels experience multiple loading

conditions during the operations of these vehicles. EBF3PanelOpt has the capability to optimize flat/

curved multi-sided panels with straight/curved edges having curvilinear, blade-type stiffeners under

multiple loading conditions. The mass of the panel is minimized subjected to constraints on buckling,

von Mises stress, and crippling or local failure of the stiffener using global optimization techniques or

gradient based optimization techniques. The panel/stiffener geometry is parametrized using design

variables that include variables for orientation and shape of the stiffeners, the thicknesses and height of

the stiffeners, and the plate thickness. The plate can have uniform thickness or non-uniform thicknesses

for the pockets created by the stiffeners. During optimization, constraints can be applied for each of the

loading conditions by aggregating all the responses using Kreisselmeir–Steinhauser criteria or using

worst response amongst all the responses or applying all the constraints. Initially, the flat rectangular

panel is optimized for the single load-case to study the effectiveness of the panel thickness option.

Then, the optimization of flat rectangular and cylindrical panels is carried out for three sample load

cases of practical interest. This paper discusses the advantages and disadvantages of the proposed

constraints’ application.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The dream to build robust, multi-functional, fuel-efficient andenvironment friendly (Green) aerospace vehicles is becomingpossible due to the advancements in numerical methods likefinite element method and computational fluid dynamics; andadvances in optimization algorithms along with computer-aidedmodeling, computer-aided machining and the availability of everincreasing computational power. Unstable fuel prices have led usto increase the fuel-efficiency which in-turn demands thesedesigns to be lighter. Reduced delivery time to customers forcedaircraft industries to have minimal lead and tooling times for

ll rights reserved.

: þ1 540 231 9632.

K. Kapania).

nia.html (R.K. Kapania).

manufacturing these vehicles. Hence, it leads to monolithicconstruction of vehicles with the minimum possible number ofpart-count. Manufacturing these structures with complex shapesusing non-traditional techniques may yield lighter and betterdesigns. The recent monolithic construction of unmanned air-vehicle can be found at http://www.asminternational.org/portal/site/www/NewsItem/?vgnextoid=19168aee5e071310VgnVCM10-0000621e010aRCRD. Boeing [1] developed an integrally stiffenedfuselage concept, for which analysis and experimental testsdemonstrated equal or better performance compared with con-ventionally fabricated structures, with regard to weight andstructural integrity, while achieving a significant reduction inmanufacturing cost.

Traditional wing and fuselage are designed as a ‘Semi-mono-coque’ structure where internal structure resists external loads andskin only acts as an external covering. Traditional wings havestraight spars and ribs which are attached to the skin panels usingspar and ribs caps using riveted joints. Most of the times, these skin,

S.B. Mulani et al. / Thin-Walled Structures 63 (2013) 13–2614

spar and rib panels have uniform thickness. Skin panels are usuallystiffened with straight stringers to lower their weights. Fuselageskeleton is created using ring frames and longerons; and fuselageskin panels are stiffened with straight stiffeners, if required. The useof straight profile and uniform thickness for spars, ribs, stiffenersand skin limits the optimization design space. This design space canbe increased by employing curved, non-uniform thickness stiffeningmembers as well as panel thicknesses without using riveted joints. Amonolithic construction of the vehicle and a reduced count of totalparts for an assembly lead to the concept of Unitized Structureswhich might use curvilinear stiffening members [2]. The benefits ofunitized structures [3] can be summarized as follows: (a) reducedpart count, manufacturing cycle time, and fabrication cost; (b) addeddesign flexibility, weight savings, inspectability, and resistance tofatigue and corrosion; (c) enhanced automation, improved ergo-nomics, and reduced work; and (d) increased determinant assemblyopportunities, improved fit, and reduced rework.

Innovative manufacturing techniques like Rapid manufactur-ing, Rapid prototyping, and Solid Freeform Fabrication (SFF), inwhich material is deposited instead of removing from a block ofmaterial to build complex structures [4], are in development.Friction Stir Welding (FSW) [5] and Electron Beam Free FormFabrication [6,7] (EBF3) would become especially useful in aero-space industries because these techniques can be used withaluminum and titanium alloys. Using these techniques, thearbitrarily curved stiffened panels which are low-cost high-performance panels and have a grid of stiffeners with a prescribeduniform or non-uniform stiffening pattern, such as isogrid andgeodesically stiffened panels [8,9], can be manufactured for futureaerospace vehicles.

Bedair [10] pointed out that the size of the stiffener influencesits optimal location; hence the stability of the stiffened panel is afunction of stiffener size and locations. So, the optimization ofstiffened panel where both the stiffening members’ locations andsize are design variables has multiple optimal solutions [11–14].The recent work on the optimization of the aircraft panel, withfixed locations of the stiffeners but stiffener cross-section mem-ber dimensions are the design variables, has multiple minima[15]. The optimization of panel with curvilinear stiffeners andstiffener locations, curvature and size as design variables neces-sitates the global optimization techniques. Genetic Algorithms(GA), Particle Swarm Optimization (PSO), and Response SurfaceApproaches (RSA) which are computationally expensive as com-pared to gradient based optimization could be utilized to opti-mize the curvilinear stiffened panel. Due to advances incomputational power and resources and the cost-effectivenessof these resources, one can utilize coarse-grained parallelism toobtain a better optimal solution.

Hypersizer [16] is the only commercial software that opti-mizes stiffened panels with different configurations of straightstiffeners using semi-analytical and finite element analyses.PASCO [17], PANDA [18], and VICONOPT [19], government-proprietary frameworks optimize the stiffened panels using adiscrete number of straight stiffeners for uniform in-plane loadswhich internally use the Ritz or finite strip method to perform theanalysis. But these frameworks cannot be used with non-uniformloading and non-rectangular panels; to read further about thecritical review of these frameworks, see [19,20]. Advanced com-puting, optimization algorithms and multi-physics simulationsusing numerical methods have enabled us to design and optimizeaerospace vehicles using unitized structures. So, an optimizationframework is built using MD-Patran2 (Geometry Modeling),

2 MD-Patran 2008 r1, MSC Software Corporation, Santa Ana, California.

MD-Nastran3 (FEM Analysis), and Python4 (Central Processor) tooptimize the stiffened panels using curvilinear blade stiffeners.

The remainder of this paper is arranged as follows. In Section 2,the developed optimization framework for stiffened panel isdescribed with brief description of the PSO technique. The proposedframework can be utilized to optimize the curvilinearly blade-stiffened panels using GA, RSA, and gradient-based approaches,but here, PSO is employed. Section 3 discusses the optimizationframework with regard to two critical processes: (1) generation ofthe Patran initial database which stores geometry of the panel,loads, boundary conditions and materials and (2) parametricstiffener geometry modeling. The optimization problem isdefined in Section 4 wherein the stiffener locations and the sizingoptimization variables are described; the stiffened panelresponses are also discussed in the same section along with thethree different techniques to apply these constraints for multipleload cases. Initially, a flat panel is optimized for single load-casewith different options for panel thickness and the results arepresented in Section 5. The optimization of flat and cylindricalpanels for multiple-load conditions is shown in the same section.Finally, conclusions are summarized.

2. Optimization framework

Mathematically, the optimization problem having continuousdesign variables is defined as

minx

f ðxÞ

giðxÞr1, i¼ 1, . . . ,m

ajrxjrbj, j¼ 1, . . . ,n ð1Þ

Even though, in traditional manufacturing, design variables arediscrete; generally, the optimization problem is defined usingcontinuous variables because their use makes the optimizationproblem to be more tractable and efficient. For SFF manufacturinglike EBF,3 all sizing design variables are continuous, so continuousvariables are used during optimization in this study. In Eq. (1),f ðxÞ is the objective function which is to be minimized withrespect to the design variables, x, while satisfying the constraints,giðxÞ. The design variables are defined using lower, aj, and upper,bj, limits/bounds. As described earlier, stiffened panel optimiza-tion has multiple minima especially when the stiffener size, theorientation and the location are the design variables of choice[11,13,15].

Kapania et al. [12,14] have shown that a flat stiffened panelwith curvilinear stiffeners has a design space that has multipleminima. To obtain full advantage of unitized structures and todeal with a discontinuous design space, PSO is utilized to mini-mize the weight of the stiffened panel. One can also use GA or RSAor gradient-based optimization instead of PSO to optimize thestiffened panel with curvilinear stiffeners. Here, PSO is describedin short, for detailed information on PSO, see [21]. The PSOalgorithm is described in Fig. 1. In brief, random particles(designs) are distributed and evaluated. These particles’ designvariables are updated based upon individual and social correc-tions. In PSO, social correction and individual correction areobtained meaning that the whole random particles group(swarm) direction and individual particle’s direction during opti-mization are updated, respectively. For calculating new designvariables, Eq. (3) is used during PSO. Design variables are calledthe positions of the particles and are denoted as xk

i as given inEq. (3), where vk

i is the velocity of the particle, w is the inertia

3 MD-Nastran R2.1 MSC Software Corporation, Santa Ana, California.4 Python 2.6.4, www.python.org.

S.B. Mulani et al. / Thin-Walled Structures 63 (2013) 13–26 15

weighting parameter of velocity, c1 and c2 are the thrust para-meters, r1 and r2 are the uniform random numbers. The para-meters, pi and pk

g, are the best particle position and the bestswarm position, respectively. Generally, time step, Dt is taken as aunit time. The inertia weight parameter, w, decides the influenceof particle’s velocity as compared to personal and social influ-ences. This parameter decides optimization convergence rate andhence the optimal solution

vikþ1 ¼wvi

kþc1r1ðpi�xi

kÞ

Dtþc2r2

ðpgk�xi

kÞ

Dtð2Þ

xikþ1 ¼ xi

kþvikþ1Dt ð3Þ

The optimization framework, EBF3PanelOpt shown in Fig. 2 isdeveloped using an object oriented script written in Python thatinterfaces MD-Patran and MD-Nastran to perform FEA on a panelwith curvilinear, blade stiffeners and returns the mass of thepanel and constraints on yielding, buckling, and crippling or localfailure of the panel. In this framework, the user needs to create aninitial Patran database using either EBF3PanelOpt or utilizingPatran directly [22]. The creation of the initial Patran databaseusing EBF3PanelOpt supports four-edged panel geometry, uniform in-plane and applied transverse loads, and boundary conditions formultiple load cases. The idea behind creating the initial Patrandatabase is that the entities that are not changing during optimizationshould be created only once. In the EBF3PanelOpt framework, theinput data that is not changing during the optimization is written inthe ASCII file called, ‘stiffened_plate_input_data.txt’. This input file

xO

xP

xG

VN

xN XO Particle’s Current PositionXN Particle’s New PositionVN Particle’s New VelocityXG Swarm’s Best PositionXP Particle’s Best Position

Design Space

Fig. 1. Algorithm of particle swarm optimization process.

Fig. 2. EBF3PanelOpt optimization fram

contains the definition of in-plane loads and boundary conditions forall applied load-cases, the material definition, average triangularmesh size, Kreisselmeir–Steinhauser criteria constant, and eightpoints which define the geometry of the panel. By specifying thisinformation, the user is making sure that this data is not changing butis utilized in the optimization. The design variables are written in‘dvar.vef’ which would be changing during the optimization. If theuser wants to constrain some of the design variables, ‘dvar.vef’should be split in such a way that it contains both sets of designvariables, the changing ones and that must stay constant during theoptimization.

Efforts are also going on to apply optimization to designvehicle using multi-level optimization. Hence, it would be advi-sable to utilize FEM to design full scale vehicle, at least on thecomponent level (wing, fuselage). Once, the FEM definition of thevehicle or its components (may be with a coarser mesh) isavailable, the panel optimization is easier. For the local optimiza-tion (panel optimization), first, the panels between stiffeningmembers (ribs and spars of wing, and longerons and ring framesof fuselage) would be extracted from the global model specifyingthe number of groups of elements and nodes related to eachpanel. The corresponding element properties such as materialproperties and thickness are transferred automatically to the localpanel using MD-Patran input session file. Once the FEM models oflocal panels are created, the displacement and stress fields fromthe global FEM model solution could be used as load cases andboundary conditions for the local panel analysis. Then, the userwould decide which panels need to be optimized so the rest of theglobal structure would not be changing. Using the definition ofthe optimized panels, the global model (vehicle or component)should be updated and the user can carry global analysis of wholevehicle. During the local optimization of panels, local meshrefinement could be carried and super-element definition of theoptimized panel could be utilized to represent panels in theglobal model.

The nature of the optimization task is such that there willalways be some geometries for which Patran cannot meshproperly. For these geometries, it is essential that EBF3PanelOpt

can recognize the failure and take appropriate action to informthe optimizer that a response was not returned. EBF3PanelOpt

ework for blade-stiffened panel.

VisualDoc/DAKOTA

Does the design make sense?

Patran

design.out

No Yes

Did the session file finish?

No Yes

Read stiffener intersections

Patran

Did the session file finish and is the geometry

meshedproperly?

No Yes

(1) Analysis Failed.

Return failed responses.

Call NastranFEA. Return responses.

(see 1)

(see 1) Play Patran session file 1.

Play Patran session file 2.

Fig. 3. Error mitigation in EBF3PanelOpt framework (Yellow indicates EBF3PanelOpt operations. Blue are external interactions). (For interpretation of the references to color

in this figure legend, the reader is referred to the web version of this article.)

x1

x4

(x2, x3)ControlPolygon

Fig. 4. Design variables identifying the stiffener shape and position.

S.B. Mulani et al. / Thin-Walled Structures 63 (2013) 13–2616

recognizes the following failure types: the Patran session file didnot finish playing (likely because of difficulty in meshing),the mesh paver failed, and the design variables do not makephysical sense such as when a stiffener begins and ends at thesame point. A flowchart for the avoidance of these three failuretypes is shown in Fig. 3.

EBF3PanelOpt then runs a Patran session file which creates thegeometry and determines if any stiffeners are intersecting. Nomeshing is done in this step. When Patran is executed, a filenamed myses.ses.01.myR.01 is recorded which contains all theinformation Patran returned while running the session file.EBF3PanelOpt reads this file to check that the session file ranproperly and if there were any intersecting stiffeners. Next,EBF3PanelOpt calls Patran again to associate the intersectingstiffeners so that the mesh connectivity at the stiffener intersec-tion will bind the stiffeners together and facilitate load transferdirectly between the intersecting stiffeners. A Nastran input filewith the FEM details is written.

During the second Patran execution, EBF3PanelOpt reads therecorded myses.ses.01.myR.01 file to verify the successful generationof the mesh. If all steps were successful, then the FEA is performedand the responses are returned. If there was an error detected alongthe way, the pass/fail response indicates a failure so that the designcan be disregarded without discontinuities being introduced into thedesign space. During the analysis with ‘Non-Uniform Panel Thickness’option, the plate pockets formed due to stiffeners may have differentthicknesses as compared to one thickness for the whole plate. Asorting algorithm finds out the appropriate panel-pocket edges andtheir association with original panel edges before the creation of thepockets and applies boundary conditions and loads to these pockets[22]. ‘Non-Uniform Panel Thickness’ option increases the number ofdesign variables and the user has to provide more design variables forpockets as compared to the required number for the safety of theoptimization process.

3. Curvilinearly blade-stiffened panel parametric geometrygeneration

A key feature of EBF3PanelOpt is the ability to specify thegeometry in a parametric fashion such that the optimizer fully

specifies the panel shape and size. Sizing quantities such as panelthickness, stiffener thickness, and stiffener height are used asdesign variables to define the geometry. The stiffener curve isrepresented using third order uniform rational B-spline using twoend-points and a control point so the stiffener always remains inthe panel area. One can use more points to define the stiffenercurve, but it increases the number of design variables, also. As thestiffener end-points lie on the perimeter of the panel, a singleperimeter curve is created in Patran. The end-points are createdusing parametric extraction of the perimeter curve. The end-pointdesign variable always lies between zero and one. The control-point is defined using interpolation of the panel surface, so thecontrol-point’s x co-ordinate and y co-ordinate have valuesbetween zero and one. So, a panel with arbitrary geometry, alongwith the stiffeners, can be represented using the parametrization.

In Fig. 4, the design variables identifying the shape andposition of the stiffener are indicated. Note that the end-pointsof each stiffener are indicated by design variables x1 and x4, andcontrol-point is having design variables, x2 and x3. For uniformcross-section blade-stiffener, height and thickness are designvariables apart from plate thickness as a design variable. Cur-rently, EBF3PanelOpt supports any number of curvilinear blade-stiffeners, but it is recommended that maximum six stiffeners

S.B. Mulani et al. / Thin-Walled Structures 63 (2013) 13–26 17

should be utilized during the optimization. The minimum numberof design variables would become six times the number ofstiffeners, in addition to the design variables for plate.For example, three and five stiffeners’ case, the stiffened panelhaving uniform panel thickness has 19 and 31 design variables,respectively. If the ‘Non-uniform Panel Thickness’ option isemployed for the optimization, then the number of design ofvariables would be 25 and 40 for the case of three and fivestiffeners’ panel due to additional design variables for panelpockets thicknesses.

4. Constraints calculation and optimization problemformulation

The optimization goal is to minimize the mass of the stiffenedpanel, whereas constraints are imposed on global buckling,von Mises stress and crippling or local failure of the stiffeners.The linear buckling analysis is carried out using the SOL-105solver from MD-Nastran that gives the lowest buckling eigenva-lue or the critical buckling load factor as well as von Mises stressdistribution for the static analysis. Nastran’s ‘SUBCASE’ and‘STATSUB’ utilities are used to apply multiple load cases. Thesuccessful execution of this Nastran (bdf) file, the Nastranresponse file (f06) is read by Python. Displacements, elementalstresses and forces are read from f06 file to obtain responses likebucking factor, von Mises stress and crippling stress using Python.Crippling criteria for the stiffeners is calculated using the for-mulation given by Niu [23]. After obtaining these differentresponses, they are normalized to process in the optimizationproblem. The buckling factor, BF0 constraint, inverse of thefundamental eigenvalue, l0, is calculated using Eq. (4). Thefundamental eigenvalue, l0, the normalized load with respect tothe applied load represents the buckling instability when the loadapplied. l0 greater than one implies positive margin of safety

BF0 ¼1

l0r1 ð4Þ

The von Mises stress, svm, in the stiffened panel should be lessthan the given safe stress criteria during optimization. Ideally,constraint on stress in each element should be specified. How-ever, the FE mesh topology is changing for every design due to theinclusion of shape design variables. In that case, either themaximum constraint value or aggregated constraint value overthe entire FE mesh can be used. Using maximum constraint valuemight display a discontinuous behavior. The von Mises stressesfor all elements aggregated using Kreisselmeier and Steinhauser[24] (KS) criteria and the constraint is imposed on KSs as follows:

KSsðsÞ ¼1

r ln1PN

i ¼ 1 Ai

XN

i ¼ 1

Aiersvmi

=sy

!ð5Þ

here r is a constant whose value changes the behavior of theconstraint, for example, a high r value tends to direct theconstraint value towards a maximum and a low r value tendsto smooth the stresses. In the present implementation, a ratherhigh value, 150, is used. Additionally, to avoid stress concentra-tions localized in very small areas, a weighted average is takenusing individual element areas (Ai) of individual elements in theFE mesh. For some cases, si=sy44:6, the KS function cannot beevaluated using double precision arithmetic. When this situationis encountered, the maximum constraint value was taken and thediscontinuities accepted. Such a design is obviously poor and thediscontinuities are not likely to occur in the explored areas of thedesign space. This can be avoided by proper choice of designvariable bounds. The KS criteria is conservative, however itseffectiveness in optimization has been proved with thousands of

constraints [25]. The crippling constraint is calculated by

ss

Fccr1 ð6Þ

where ss is the maximum negative principal stress in the stiffenerand Fcc is the maximum allowable stress for blade stiffener and iscalculated using the formula

Fcc ¼

sy, 0:61525bffiffiffiffiffiffiffiffiffiffiffisy=E

pt

!�0:78387

41

0:61525sybffiffiffiffiffiffiffiffiffiffiffisy=E

pt

!�0:78387

, 0:61525bffiffiffiffiffiffiffiffiffiffiffisy=E

pt

!�0:78387

r1

8>>>>>><>>>>>>:

ð7Þ

where E is Young’s modulus of the stiffener material, b is thestiffener height, t is the stiffener thickness. The maximum allow-able stress formula, Eq. (7), was obtained from the plot given in[23, p. 444]. The given plot is applicable for general extrudedsections and for any material. Hence, Eq. (7) applicability is validfor rapid manufacturing techniques like FSW or EBF3. So, theoptimization problem for a stiffened panel for the single load casecan be written as

minx

massðxÞ

BF0r1

KSsr1ss

Fccr1 ð8Þ

The EBF3PanelOpt framework supports the optimization ofstiffened panels subjected to multiple load cases. To do this, theload cases are generated through Patran a priori in the optimiza-tion loop. Three options are provided to the user for constraintaggregation for the multiple load cases. These are (1) All

Responses, which print the constraints for each load case, (2)Worst Responses, which print the maximum constraints, or (3) theKS Aggregation, which uses the KS function to create a singlecontinuous constraint. Each of these approaches has merits anddemerits. Approach (1) provides continuous constraints; how-ever, the number of constraints increases with the number of loadcases. Approach (2) provides discontinuous constraints; however,despite the number of load cases, there is always a constantnumber of constraints. Approach (3) provides continuous con-straints and the number of constraints does not increase with theincreasing number of load cases; however, the KS function willfail if the constraint is very large (44:5 with the KS parameterr¼ 150; in such a case, we switch to the maximum worstconstraint case and accept the discontinuities). Thus, there is noclear best choice for constraint aggregation, but one can utilizethe recently updated KS function definition that avoids numericaldifficulties [26].

5. Numerical examples

The developed EBF3PanelOpt is applied to obtain optimaldesigns for the load cases and examples suggested by LockheedMartin Aeronautics Company. For all examples presented here,the ultimate loads are obtained after applying a factor of safety of1.5 to the limit loads and the panels are designed for ultimateloads. Initially, the flat rectangular panel which is part of a wingpylon rib is optimized for the single compression dominant load-case using both the uniform and the non-uniform panel thicknessoptions in EBF3PanelOpt. The rib panel edges are fixed meaningthat while the rotations and transverse displacements are notallowed, the in-plane displacements are at these edges. The ribpanel is optimized using four blade stiffeners. Afterwards,

Table 1Material properties for Al 2139.

Young’s modulus (GPa) 73.085

Poisson’s ratio 0.33

Yield stress (MPa) 427.47

Density (kg/m3) 2700.00

Table 2VisualDOC optimization parameters for PSO optimization.

Inertia weight parameter 1.40

Local optimum trust parameter, c1 1.50

Global optimum trust parameter, c2 2.50

Absolute objective convergence 1E�06

Relative objective convergence 1E�03

Absolute design variable convergence 1E�06

Relative design variable convergence 1E�03

Table 3The sizing design variables’ constraints.

Design variables Lower bound (m) Upper bound (m)

Stiffener height 0.01 0.05

Stiffener thickness 0.001 0.008

Plate thickness 0.001 0.006

X

Y

w, rx = 0

w, ry = 0

u, v = 0u = 0

w, rx = 0

w, r

y =

0

NXX

NYY

NXY

0.7112 m

0.60

96 m

Fig. 5. Transport aircraft wing Pylon rib panel geometry along with the loads.

Table 4Pylon wing panel optimization results.

Responses/Variables

UniformPSOþGBO

Uniform PSOþNon-uniform GBO

Non-uniformPSOþGBO

Mass (kg) 3.2337 3.1896 3.0732

Buckling factor 1.0015 1.0001 1.0001

KSC 0.2109 0.2151 0.2958

CrippCon 0.9137 0.9369 0.7087

PSO iterations 302 302 358

GBO iterations 5 14 0

Stiffener1 height (m)

4:8865E�02 4:8865E�02 2:2914E�02

Stiffener2 height (m)

4:8993E�02 4:8995E�02 2:6180E�02

Stiffener3 height (m)

1:3403E�02 1:3333E�02 4:8141E�02

Stiffener4 height (m)

2:9993E�02 2:9995E�02 2:6180E�02

Stiffener1 thickness (m)

1:0003E�03 1:0010E�03 3:3866E�03

Stiffener2 thickness (m)

1:1290E�03 1:1004E�03 2:3925E�03

Stiffener3 thickness (m)

2:5810E�03 2:3629E�03 3:3866E�03

Stiffener4 thickness (m)

1:0002E�03 1:0285E�03 2:3925E�03

Minimum andmaximum

2:4502E�03 2:3847E�03 1:6896E�03

Plate thickness(m)

2:6757E�03 2:7839E�03

S.B. Mulani et al. / Thin-Walled Structures 63 (2013) 13–2618

the panels with multiple load-cases are optimized to measure theeffectiveness of the constraints aggregation. Lockheed Martinsuggested two sets of examples: (1) Two flat rectangular panels,and (2) a cylindrical panel for the multiple load-cases analysis.The material used for these panels is the aluminum alloy, Al 2139;its properties are given in Table 1. All the panels with multipleload-cases are simply supported, meaning that normal/transversedisplacements are restrained, and rotations are allowed. In-planedisplacements are allowed and two corner points are restrainedsuch that rigid-body motions would be avoided. All three panels,with multiple load-cases, are designed using two uniform cross-section blade-stiffeners.

During mesh generation of stiffened panels using MD-Patran,the mesh size parameter is set to be 0.01, meaning that theaverage elemental edge length would be 0.01 m, approximately,and MD-Nastran’s ‘CTRIA3’ triangular first order shear deformableshell elements are used. VisualDOC optimization parameters forPSO optimization are given in Table 2. The maximum number ofiterations are set to be 800 and 24 particles are used in PSO. AfterPSO optimization, gradient-based optimization is carried out torefine the designs. During the optimization, a constraint can beviolated by no more than 0.3%. The optimization convergenceparameters for design variables and objective are the same forboth PSO and gradient-based optimization. The sizing designvariables include the panel thickness, the heights of the stiffenersand the thicknesses of the stiffeners, and the minimum and themaximum values of the sizing variables are given in Table 3. Theend-points of the stiffeners are allowed to be anywhere on thewhole perimeter of the panel to compare the different constraintstrategies.

5.1. A transport aircraft wing pylon rib panel

The developed optimization framework is applied to obtainoptimal designs of a flat rectangular panel, having dimensions,0.6096 m�0.7112 m shown in Fig. 5. This panel is representative

of a large wing engine pylon rib and is optimized for minimummass. The panel is fixed meaning that rotations and transversedisplacements are not allowed, but in-plane displacements areallowed. The rib panel is optimized using four blade stiffeners forthe single compression dominant load-case using both the uni-form and the non-uniform panel thickness options. Here,NXX ¼�140:28 kN=m and NYY ¼�44:307 kN=m are applied in-plane axial loads, and NXY ¼ 15:761 kN=m is applied in-planeshear load. Positive axial loads indicate tension. The panel isoptimized using different options for panel thickness as three casestudies: (1) PSO and GBO with uniform panel thickness, (2) PSOwith uniform panel thickness, but GBO with non-uniform panelthickness, and (3) Both PSO and GBO with non-uniform panelthickness. Here and in the rest of the paper, GBO refers togradient-based optimization. For these case-studies, optimalresults are given in Table 4. The stiffener heights and thethicknesses for ‘Uniform PSOþGBO’, ‘Uniform PSOþNon-uniform

Fig. 6. Pylon wing panel optimization using uniform PSO and GBO panel thickness: (a) von Mises stress distribution and (b) first buckling mode shape.

Fig. 7. Pylon wing panel optimization using uniform PSO and non-uniform GBO panel thickness: (a) von Mises stress distribution and (b) first buckling mode shape.

Fig. 8. Pylon wing panel optimization non-uniform PSO and GBO panel thickness: (a) von Mises stress distribution and (b) first buckling mode shape.

S.B. Mulani et al. / Thin-Walled Structures 63 (2013) 13–26 19

GBO’, and ‘Non-uniform PSOþGBO’ are provided in Table 4 alongwith the minimum and maximum panel pockets’ thicknesses. KSC

and CrippCon in Table 4 and the rest of the paper represent KScriteria of von Mises stresses of the panel and crippling criteria forthe stiffeners. The von Mises stress distribution and first bucklingmode shape are shown in Figs. 6, 7, and 8 for case-studies:(1) Uniform panel thickness PSO and GBO, (2) Uniform panelthickness PSO and Non-Uniform panel thickness GBO and(3) Non-uniform panel thickness PSO and GBO, respectively. The‘Non-uniform Panel Thickness’ option has reduced the massconsiderably. Depending upon the available computationalresources, the user can employ ‘Non-uniform Panel Thickness’with global optimization techniques or with GBO. The number of

local pockets for buckling mode shape is increased from three orfour for ‘Uniform PSOþGBO’, to six for ‘Uniform PSOþNon-uni-form GBO’ and to eight for ‘Non-uniform PSOþGBO’ thicknessoptions for optimization of rib panel and subsequently mass isdecreased. KSC criteria is increased for these thickness optionsmeaning that panel is getting more stressed for better optimaldesigns in the order of ‘Uniform PSOþGBO’, ‘Uniform PSO þ Non-uniform GBO’, and ‘Non-uniform PSOþGBO’.

5.2. A conventional aircraft wing skin flat panel

The optimization is carried out for the rectangular panelhaving dimensions, 0.4064 m�0.5080 m as shown in Fig. 9 using

S.B. Mulani et al. / Thin-Walled Structures 63 (2013) 13–2620

three different constraint aggregation approaches for three loadcases as given in Table 5. This panel is of upper wing skin from theroot section. Here, NXX and NYY are applied in-plane axial loads,and NXY is applied in-plane shear load. This panel is optimizedusing PSO optimization and the results are compared for threedifferent constraints’ aggregation approaches in Table 6. In

X

Y

w = 0

w = 0

u, v = 0u = 0

w = 0

w =

0 NXX

NYY

NXY

0.5080 m

0.40

64 m

Fig. 9. Conventional aircraft wing panel geometry along with the loads.

Table 5Conventional aircraft wing panel load cases.

Case NXX ðkN=mÞ NYY ðkN=mÞ NXY ðkN=mÞ

L1 �1145.5 �51.32 75.48

L2 �1023.1 �43.79 106.66

L3 650.0 21.55 �133.49

Table 6Conventional aircraft wing panel optimization results.

Responses/Variables

AllResponses

WorstResponses

KSAggregation

Mass (kg) 3.5356 3.4603 3.6046

Buckling factor 1:0023 ðL1Þ 1:0026 ðL1Þ 1:0006 ðL1Þ

KSC 0:5188 ðL1Þ 0:5330 ðL1Þ 0:4970 ðL1Þ

CrippCon 0:8276 ðL1Þ 0:6979 ðL1Þ 0:6392 ðL1Þ

PSO iterations 800 800 512

GBO iterations 5 4 5

Stiffener 1 height (m) 1:7936E�02 3:7338E�02 4:8141E�02

Stiffener 2 height (m) 3:9090E�02 4:9990E�02 2:6180E�02

Stiffener 1 thickness (m) 3:1159E�03 4:5204E�03 3:3866E�03

Stiffener 2 thickness (m) 4:9948E�03 3:5282E�03 2:3925E�03

Plate thickness (m) 5:8039E�03 5:3281E�03 5:8842E�03

Fig. 10. Conventional aircraft wing panel optimization using All Responses constrain

Table 6, worst constraints amongst load-cases are indicated inparentheses in front of the constraints. In Figs. 10, 11, and 12, vonMises stress distribution due to load-case, L1, and buckling modeshapes due to load-case, L1, of optimal designs are shown usingdifferent constraint aggregations. All optimal designs are detectedby load-case, L1. Only one stiffener along the major compressiondirection is taking part in the load-sharing phenomenon as seenfrom Fig. 10 for the case of All Responses constraint strategy. Forthis panel, Worst Responses constraint aggregation gives muchbetter optimal design as compared to other constraint aggrega-tion techniques and both stiffeners are playing role in the load-sharing phenomenon, so panel thickness is smaller as comparedto other constraint techniques’ optimal designs.

5.3. A blended body wing flat panel

The panel is from the junction of outer wing and body of theBlended-Body-Wing aircraft configuration. The dimensions of thepanel and load conventions are the same as given in Section 5.2.The panel is optimized using the load cases given in Table 7 anddesigned using two blade-stiffeners. The optimization results forthis panel using different constraint aggregation techniques aregiven in Table 8. Optimal designs are governed by B2 and B3 load-cases which are compression dominant. As shown in Table 8, KS

Aggregation constraint strategy results in better optimal design ascompared to other constraint aggregations. The von Mises stressdistribution due to the load-case B2 and first buckling modeshapes due to the load-case B3 are shown in Figs. 13, 14, and 15for All Responses, Worst Response and KS Aggregation constraintaggregation strategies, respectively. Only one stiffener is effectivein load-sharing in the case of optimal design of All Responses

constraint strategy as shown in Fig. 13. In the case of Worst

Responses constraint strategy, both the buckling factor and thevon Mises stress constraints are active as shown in Table 8. Forthis panel, KS Aggregation results in a lower optimal mass ascompared to using other constraint aggregation strategies.

5.4. A cylindrical fuselage panel

The cylindrical panel shown in Fig. 16 is a part of lower fuselageskin and has a radius of curvature, 1.7272 m, length, 0.5080 m andcircumferential length, 0.4064 m. The panel is optimized for the loadcases given in Table 9. Here, NZZ and NTT are applied in-plane axialloads along the Z direction and the tangential direction, respectively,and NRT is applied in-plane shear load. During the application ofmaximum tension load case, outward radial pressure, 45.161 KPa isapplied and the developed hoop tension and axial tension due to theapplied pressure are also added to tension load case. The optimizationresults are compared for different constraint strategies in Table 10.The result of the Maximum Response case is better than those from the

t strategy: (a) von Mises stress distribution and (b) first buckling mode shape.

Fig. 11. Conventional aircraft wing panel optimization using Worst Response constraint strategy: (a) von Mises stress distribution and (b) first buckling mode shape.

Fig. 12. Conventional aircraft wing panel optimization using KS Aggregation constraint strategy: (a) von Mises stress distribution and (b) first buckling mode shape.

Table 7Blended body wing panel load cases.

Case NXX ðkN=mÞ NYY ðkN=mÞ NXY ðkN=mÞ

B1 233.1 263.22 599.29

B2 �275.83 531.69 635.36

B3 �280.38 �90.02 �234.50

Table 8Blended body wing panel optimization results.

Responses/Variables

AllResponses

WorstResponses

KSAggregation

Mass (kg) 2.5511 2.4992 2.4170

Buckling factor 1:0029 ðB3Þ 1:0020 ðB3Þ 1:0027 ðB3Þ

KSC 0:7870 ðB2Þ 1:0029 ðB2Þ 0:8646 ðB2Þ

CrippCon 0:5469 ðB2Þ 0:8705 ðB3Þ 0:4920 ðB2Þ

PSO iterations 202 800 512

GBO iterations 7 11 6

Stiffener 1 height (m) 1:4809E�02 2:5762E�02 4:9966E�02

Stiffener 2 height (m) 3:9666E�02 4:0318E�02 1:0441E�02

Stiffener 1 thickness (m) 2:1314E�03 6:6155E�03 1:5743E�03

Stiffener 2 thickness (m) 4:3960E�03 2:1148E�03 7:9991E�03

Plate thickness (m) 4:0770E�03 3:7377E�03 3:8798E�03

S.B. Mulani et al. / Thin-Walled Structures 63 (2013) 13–26 21

other two constraint aggregation strategies. The von Mises stressdistribution and first buckling mode shape due to the load-case L1 areshown in Figs. 17, 18, and 19 for All Response, Worst Response and KS

Aggregation constraint aggregation strategies, respectively.From Table 10, it can be seen that the optimal design is

governed by load-case, L1 which is compression-dominant forall constraint strategies. So the cylindrical panel is optimized forL1 load-case with different option for panel thickness. Three case

studies are carried out: (1) PSO and GBO with uniform panelthickness, (2) PSO with uniform panel thickness, but GBO withnon-uniform panel thickness, and (3) Both PSO and GBO withnon-uniform panel thickness. For these case-studies, optimalresults are given in Table 11. The von Mises stress distributionand first buckling mode shape are shown in Figs. 20, 21, and 22for case-studies: (1) Uniform panel thickness PSO and GBO,(2) Uniform panel thickness PSO and Non-Uniform panel thick-ness GBO and (3) Non-uniform panel thickness PSO and GBO,respectively. The thicknesses for ‘Uniform PSOþGBO’, ‘UniformPSOþNon-uniform GBO’, and ‘Non-uniform PSOþGBO’ thicknessoptions are compared in Fig. 23. The panel is buckling in a highermode with an increasing number of local pockets when using‘Uniform PSOþGBO’, ‘Uniform PSOþNon-uniform GBO’, and‘Non-uniform PSOþGBO’ thickness option, and with adecreasing mass.

6. Conclusion and future work

The optimization framework, EBF3PanelOpt has two optionsfor selecting panel thickness: (1) ‘Uniform Panel Thickness’and (2) ‘Non-uniform Panel Thickness’. These options can beused with global optimization techniques or gradient basedoptimization depending upon the computational resources.Using ‘Non-uniform Panel Thickness’, optimal mass of thepanel can be reduced considerably and the panel would havehigher order buckling mode shape meaning that panel buck-ling mode shape has more panel pockets. ‘Non-uniform PanelThickness’ also increases the magnitude of averaged stressesin the panel.

EBF3PanelOpt is supported with multiple load-cases analysisand optimization of panels stiffened with curvilinear stiffenersusing different constraint aggregation strategies, simultaneously.Stiffener locations and curvature play an important role in

Fig. 13. Blended body wing panel optimization using All Responses constraint strategy: (a) von Mises stress distribution and (b) first buckling mode shape.

Fig. 14. Blended body wing panel optimization using Worst Response constraint strategy: (a) von Mises stress distribution and (b) first buckling mode shape.

Fig. 15. Blended body wing panel optimization using KS Aggregation constraint strategy: (a) von Mises stress distribution and (b) first buckling mode shape.

R = 1.7272 m

v =0

u ,v =0

NZZ

NTT

NZT

Fig. 16. Cylindrical fuselage panel geometry along with the loads.

S.B. Mulani et al. / Thin-Walled Structures 63 (2013) 13–2622

Table 9Cylindrical fuselage panel load cases.

Case NZZ ðkN=mÞ NTT ðkN=mÞ NRT ðkN=mÞ

Compression ðL1Þ �136.78 �11.21 93.70

Tension ðL2Þ 104.90 10.33 4.38

þ39.00 þ78.00

Shear ðL3Þ �93.52 �1.58 116.46

Table 10Cylindrical fuselage panel optimization results.

Responses/Variables All Responses Worst Responses KS Aggregation

Mass (kg) 1.6746 1.6513 1.7064

Buckling factor 1:0008 ðL1Þ 1:0015 ðL1Þ 1:0027 ðL1Þ

KSC 0:2243 ðL3Þ 0:2229 ðL3Þ 0:2985 ðL3Þ

CrippCon 0:9970 ðL1Þ 0:2739 ðL1Þ 0:5356 ðL1Þ

PSO iterations 800 315 301

GBO iterations 5 8 12

Stiffener 1 height (m) 4:9879E�02 2:9331E�02 2:3838E�02

Stiffener 2 height (m) 2:8215E�02 1:7304E�02 3:5353E�02

Stiffener 1 thickness (m) 1:0620E�03 2:8400E�03 1:3350E�03

Stiffener 2 thickness (m) 2:3704E�03 2:0022E�03 1:6543E�03

Plate thickness (m) 2:6277E�03 2:5358E�03 2:7136E�03

Fig. 17. Cylindrical fuselage panel optimization using All Responses constraint strategy: (a) von Mises stress distribution and (b) first buckling mode shape.

Fig. 18. Cylindrical fuselage panel optimization using Worst Response constraint strategy: (a) von Mises stress distribution and (b) first buckling mode shape.

Fig. 19. Cylindrical fuselage panel optimization using KS aggregation constraint strategy: (a) von Mises stress distribution and (b) first buckling mode shape.

S.B. Mulani et al. / Thin-Walled Structures 63 (2013) 13–26 23

Table 11Cylindrical fuselage panel optimization results for load-case L1 .

Responses/Variables Uniform PSOþGBO Uniform PSOþNon-uniform GBO Non-uniform PSOþGBO

Mass (kg) 1.5912 1.5822 1.5846

Buckling factor 0.9974 1.0016 1.0029

KSC 0.1832 0.2006 0.2510

CrippCon 0.7019 0.8001 0.7213

PSO iterations 334 334 499

GBO iterations 0 6 0

Stiffener 1 height (m) 2:9416E�02 2:9419E�02 3:0581E�02

Stiffener 2 height (m) 2:2675E�02 2:2709E�02 2:3387E�02

Stiffener 1 thickness (m) 1:0000E�03 1:0020E�03 1:1115E�03

Stiffener 2 thickness (m) 1:5794E�03 1:6641E�03 1:0427E�03

Plate thickness (m) 2:6197E�03 2:4782E�03 1:9015E�03

2:6122E�03 3:0000E�03

2:6270E�03 2:7744E�03

2:3562E�03

Fig. 20. Cylindrical fuselage panel optimization using uniform PSO and GBO panel thickness: (a) von Mises stress distribution and (b) first buckling mode shape.

Fig. 21. Cylindrical fuselage panel optimization using uniform PSO and non-uniform GBO panel thickness: (a) von Mises stress distribution and (b) first buckling

mode shape.

Fig. 22. Cylindrical fuselage panel optimization non-uniform PSO and GBO panel thickness: (a) von Mises stress distribution and (b) first buckling mode shape.

S.B. Mulani et al. / Thin-Walled Structures 63 (2013) 13–2624

optimization. The user’s knowledge or previous experience withapplied load-cases is not necessary to sort the responses ofdifferent load-cases involved in the optimization. The user can

choose one of the constraint aggregation strategies to (1) All

Responses print the constraints for each load case, (2) Worst

Responses print the maximum constraints, or (3) KS Aggregation

Fig. 23. Cylindrical fuselage panel’s optimal thicknesses for load-case L1: (a) uniform PSOþGBO; (b) uniform PSOþnon-uniform GBO; and (c) non-uniform PSOþGBO.

S.B. Mulani et al. / Thin-Walled Structures 63 (2013) 13–26 25

uses the KS function to create a single continuous constraint.Generally, Worst Responses or KS Aggregation results in betteroptimal results. While using a global optimization technique,Worst Responses criterion is good but it cannot be used withgradient-based optimization techniques. During optimizationiterations, for complex problems, responses of different load-cases become critical. KS Aggregation constraint aggregation iscontinuous in design variable domain so it is good for gradient-based optimization.

For stiffened panels where both the sizing and the shapevariables are utilized to perform optimization, the performanceof global optimization techniques like GA, PSO, simulatedannealing, and ant-colony optimization should be compared toget better optimal designs. New efficient techniques to findbetter optimal solutions like multi-step optimization should bestudied.

Acknowledgments

The work presented here is funded under NASA Subsonic FixedWing Hybrid Body Technologies NRA Contract (NASANNL08AA02C) with Karen M. Brown Taminger as the AssociatePrincipal Investigator and Contracting Office Technical Represen-tative and Richard Keith Bird as the Contract Monitor. We thankthem for their suggestions and continuous encouragement. Theauthors would like to thank our partners in the NRA project, Mr.David Havens, Mr. Robert J. Olliffe, and Dr. Steve Engelstad, allfrom Lockheed Martin Aeronautics Company of Marietta, GA, fortheir contribution to the present research.

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