Optimization of Aircraft wing

8/13/2019 Optimization of Aircraft wing

1/13

OPTIMIZATION OF A SIMPLE AIRCRAFT WING

2001-01

Susana Anglica Falco and Alfredo Rocha de Faria

Fibraforte Engenharia, So Jos dos Campos, SP, Brazil.

R. Jos Alves dos Santos 281/306, 12230-300.e-mail: [email protected]; [email protected]

Phone: 55 12 337 1416; fax: 55 12 3937 6736

ABSTRACT

A wing-like structure consisting of spars, ribs, reinforcements and skin is optimized consideringtwo cases: (i) weight minimization, and (ii) critical load maximization. The wing carries anelliptically distributed load along the span. Positioning of spars and ribs as well as dimensions of

different parts of the structure are the design variables. Results indicate that significantimprovement in terms of objective function has been achieved through the optimization

procedures.

KEY-WORDS

Structural optimization, finite elements, aerospace designs, modal and buckling analysis


2/13


3/13


3

design process is a multi-step procedure where initial steps contemplate simplifiedconfigurations and each step inherits properties from the previous ones. Therefore, theimportance of obtaining initial optimized designs is to guarantee, or at least favor, that its best

features will pass on to subsequent steps.

The simplified aircraft wing model depicted in Fig. (1) has 6000 mm total length, 1500 mm

width and 275 mm height at the chord mid point. The material properties of machined aluminumare Young modulus 73090 N/mm2, mass density 2.7 10-6 Kg/mm3 and Poisson coefficient 0.33.

The model has 4 spars, 7 ribs, 36 rectangular skin panels and beams on the edges of all theseparts as seen in Fig. (1)

X

Y

Z

V1

Figure 1. Geometric model of the wing-like structure.

The skin panels, spars and ribs have been discretized into quadrilateral plate elements

(CQUAD4) and the beam into bar elements (CBAR). The boundary conditions are fully

clamped at one end and free at the other. The wing supports an elliptical distributed load asshown in Fig. (2) of 1.84 10-4 N/mm at the clamped end dropping to 1.66 10-5at the free end.The load is equally distributed on the upper loft surface and lower loft surface. Chordwise theload is assumed to have a constant distribution.

1.6667E-5

2.5439E-5

3.4211E-5

4.2982E-5

5.1754E-5

6.0526E-5

6.9298E-5

7.8070E-5

8.6842E-5

9.5614E-5

1.0439E-4

1.1316E-4

1.2193E-4

1.3070E-4

1.3947E-4

1.4825E-4

1.5702E-4

1.6579E-4

1.7456E-4

1.8333E-4

0. 500. 1000. 1500. 2000. 2500. 3000. 3500. 4000. 4500. 5000. 5500. 6000.Carga com forma elptica

Figure 2. Wing load distribution.


4/13


4

OPTIMIZATION PROBLEM DEFINITION

In a design optimization problem, the objective is to find the values of a set of n given design

variables x n which minimize or maximize a function f(x), denoted objective-function,

while satisfying a set of equality (hk (x)=0) as well as inequality (gr(x) 0) constraints which

define the viable region of the solution space. The constraints xlj xj xuj are called side

constraints. is the set of the real numbers (Vanderplaats, 1999).

Objective-function: f (x)

Variable Vector: xn

Constraints : hk(x) = 0 k=1,..., q (1)

gr(x) 0 r=1,..., m (2)

xlj xj xuj j = 1,..., n (3)The wing model is optimized considering two optimization problems:

PROBLEM I: Objective-function to minimize: Wing massConstraint: Maximal critical load

PROBLEM II: Objective-function to maximize: Critical load

Constraint: Wing mass

Generally stated, careful selection of the search parameters, scaling of the design variables and

selection of a good initial design are often indispensable. Furthermore, it is observed that thechoice of a representative set of design variables is a decisive factor for a successful

optimization procedure as shown in the remainder of this work.

THE CHOICE OF THE WING DESIGN VARIABLES

MSC.Nastran commercial code can simultaneously solve both member dimension (sizing) andcoordinate location (shape) optimization problems and a wide range of options are available todefine the design variables. For example, design variables may be individual member dimensions

and/or grid locations, or may be linear or nonlinear combinations of these.

The allowable shapes are defined using shape basis vectors. The engineer uses these to describehow the structure is allowed to change. The optimizer determines how much the structure canchange by modifying the design variables. It is used the Direct Input Shape method to describe

these shape basis vectors. With this method, externally generated vectors are used to defineshape basic vectors and an auxiliary model analysis provides these externally generated vectors.

A total of 43 design variables, including sizing and shape, are defined for the wing model.Tables (1) and (2) present the shape and sizing variables, respectively, used in the calculations.

Figures (3), (4) and (5) show the location of the shape, the thickness sizing and the cross sectionarea sizing variables, respectively. Observe that BEAM37 to BEAM40 comprise the entire wingspan, from root to tip. Also, BEAM30 to BEAM36 are present in all ribs. Moreover, full

geometric symmetry about XZ plane is assumed.


5/13


5

Table 1. Shape design variables definition

VARIABLES DEFINITION

N1 rib 1 position

N2 rib 2 position

N3 rib 3 position

N4 rib 4 position

N5 rib 5 position

L1 spar 1 position

L2 spar 2 position

Table 2. Sizing design variables definition

VARIABLES DEFINITION COLOR

PLA11 first plate (free end, left, Fig. 4) dark yellow

PLA12 second plate light green

PLA13 third plate dark green

PLA14 fourth plate yellow

PLA15 fifth plate light yellow

PLA16 sixth plate (clamped end) emerald green

PLA21 first plate (free end, center, Fig. 4) red

PLA22 second plate dark orange

PLA23 third plate light orange

PLA24 fourth plate light pink

PLA25 fifth plate magenta

PLA26 sixth plate (clamped end) dark pink

PLA31 first plate (free end, right, Fig. 4) celestial blue

PLA32 second plate light blue

PLA33 third plate dark blue

PLA34 fourth plate celestial dark blue

PLA35 fifth plate blue

PLA36 sixth plate (clamped end) purpleRIB1 first rib portion (left) navy blue

RIB2 second rib portion (center) dark purple

RIB3 third rib portion (right) pink

SPAR1 first spar (left, Fig. 4) yellow

SPAR2 second spar light red

SPAR3 third spar dark red

SPAR4 fourth spar (right, Fig. 4) blue

BEAM30 horizontal beam 1 red

BEAM31 horizontal beam 2 green

BEAM32 horizontal beam 3 blue

BEAM33 vertical beam 1 black




BEAM37 spanwise beam 1 orange

BEAM38 spanwise beam 2 dark blue

BEAM39 spanwise beam 3 magenta

BEAM40 spanwise beam 4 brown


6/13


6

X

Y

Z

N3

N2

N1

N4

N5

L2

L1

Figure 3. Location of the shape design variables for the wing model

Figure 4. Location of the thickness sizing design variables

BEAM36

BEAM35

BEAM34

BEAM33

BEAM32

BEAM31

BEAM37BEAM38

BEAM40

BEAM39

BEAM30

Figure 5. Location of the cross section area sizing design variables


7/13


7

PROBLEM I: MASS MINIMIZATION WITH BUCKLING CONSTRAINT

The wing model is optimized for minimum weight, considering an elastic buckling constraint.

Defining Pcr as the critical buckling load and Pa the actual applied load, the buckling problem is

to find the minimum which will trigger loss of stability, where Pcr=Pa. If the minimum is

less than 1.0 the structure has buckled. Therefore, the elastic buckling constraint requires,

effectively, the solution of an eigenvalue problem.

Five cases of increasing complexity are considered. In case I only thicknesses of spars, ribs andskin panels are design variables, i.e., the basic geometry is unchanged. Case II gives more

freedom to the structure to minimize its mass; although the spars sweep angle is fixed 0o. Ribsare allowed to change their positions in case III but they must remain parallel to the flight

direction and spars are maintained fixed. Case IV considers that both spars and ribs positionvary. Finally, case V incorporates beam cross sectional areas in the set of design variables. Allcases are summarized in table (3).

Table 3. Definition of cases

CASE I II III IV VSizing variables 25 25 25 25 36

Shape variables --- 2 (Spars position) 6 (Ribs position) 8 (Spars and ribs ) 8 (Spars and ribs)

Total variables 25 27 31 33 44

Inspection of table (4), reporting the optimization results, reveals that, for cases I-IV rib and sparthicknesses where kept as low as possible whereas in case V mass reduction is achieved chieflybecause of beam cross sectional area reduction. This suggests that the beam elements played a

crucial role in terms of load carrying capacity for cases I-IV. On the other hand, in case V, asignificant portion of the load is transferred to the panels.

Figure (6) presents the optimal designs geometry obtained. An interesting pattern is observed forcase III. The tendency of two ribs (N4 and N5) to coalesce is striking and they perhaps would

have if the side constraints were relaxed. This peculiar behavior leads to the conclusion that yetanother kind of design variables should be considered in the aircraft wing optimization, namely,

the number of structural components such as ribs or stringers.

Even tough the load carrying mechanism changes from cases I-IV to V mass reduction is

consistently achieved. In the most complex case investigated a reduction of 18% was obtained.Notice that even the simplest case results in 9% mass savings what is by itself a significant

accomplishment.


8/13


8

Table 4. Optimal results for problem I

Variable Lower

bound

Initial

value

Upper

bound

CASE

I

CASE

IICASE

IIICASE

IVCASE

V

PLA11 0.3 0.80 2.00 0.32 0.35 0.30 0.36 0.77

PLA12 0.3 0.80 2.00 0.40 0.44 0.32 0.44 0.79

PLA13 0.3 0.80 2.00 0.50 0.53 0.40 0.54 0.79

PLA14 0.3 0.80 2.00 0.56 0.63 0.52 0.64 0.79PLA15 0.3 0.80 2.00 0.66 0.71 0.61 0.74 0.79

PLA16 0.3 0.80 2.00 0.75 0.80 0.64 0.83 0.79

PLA21 0.3 0.80 2.00 0.31 0.35 0.30 0.44 0.79

PLA22 0.3 0.80 2.00 0.40 0.44 0.30 0.44 0.77

PLA23 0.3 0.80 2.00 0.49 0.53 0.39 0.53 0.79

PLA24 0.3 0.80 2.00 0.58 0.63 0.49 0.63 0.79

PLA25 0.3 0.80 2.00 0.68 0.71 0.57 0.73 0.79

PLA26 0.3 0.80 2.00 0.82 0.85 0.78 0.82 0.81

PLA31 0.3 0.80 2.00 0.32 0.35 0.30 0.37 0.78

PLA32 0.3 0.80 2.00 0.42 0.45 0.32 0.46 0.77

PLA33 0.3 0.80 2.00 0.49 0.53 0.41 0.55 0.77

PLA34 0.3 0.80 2.00 0.58 0.63 0.51 0.65 0.77PLA35 0.3 0.80 2.00 0.66 0.72 0.62 0.76 0.78

PLA36 0.3 0.80 2.00 0.76 0.81 0.63 0.85 0.76

RIB1 0.3 0.80 2.00 0.32 0.36 0.30 0.38 0.76

RIB2 0.3 0.80 2.00 0.32 0.36 0.30 0.36 0.77

RIB3 0.3 0.80 2.00 0.32 0.35 0.30 0.38 0.76

SPAR1 0.3 0.80 2.00 0.30 0.30 0.30 0.30 0.77

SPAR2 0.3 0.80 2.00 0.30 0.30 0.30 0.30 0.76

SPAR3 0.3 0.80 2.00 0.30 0.30 0.30 0.30 0.79

SPAR4 0.3 0.80 2.00 0.30 0.30 0.30 0.30 0.79

BEAM30 120.0 1256.1 1500.0 ----- ----- ----- ----- 971.00

BEAM31 120.0 1256.1 1500.0 ---- ---- ---- ---- 978.50

BEAM32 120.0 1256.1 1500.0 ---- ---- ---- ---- 982.10

BEAM33 120.0 1256.1 1500.0 ---- ---- ---- ---- 962.85BEAM34 120.0 1256.1 1500.0 ---- ---- ---- ---- 969.44

BEAM35 120.0 1256.1 1500.0 ---- ---- ---- ---- 1000.0

BEAM36 120.0 1256.1 1500.0 ----- ----- ----- ----- 963.37

BEAM37 120.0 1256.1 1500.0 ----- ----- ----- ----- 981.56

BEAM38 120.0 1256.1 1500.0 ----- ----- ----- ----- 976.97

BEAM39 120.0 1256.1 1500.0 ----- ----- ----- ----- 976.97

BEAM40 120.0 1256.1 1500.0 ----- ----- ----- ----- 977.25

N1 -1000.0 0.0 +1300.0 ---- ---- +1300.0 +50.77 +66.14

N2 -1300.0 0.0 +1300.0 ---- ---- +1298.5 +50.36 -54.33

N3 -1300.0 0.0 +1300.0 ----- ---- +1262.5 +50.36 -55.38

N4 -1300.0 0.0 +1300.0 ----- ---- +1029.3 +50.36 +4.15

N5 -1300.0 0.0 +1300.0 ----- ---- +122.39 +24.99 +63.39L1 -400.0 100.0 +400.0 ----- +88.46 --- +1.24 +11.82

L2 -100.0 100.0 +100.0 ----- -38.91 --- +0.84 0.00

Mass (Kg) Initial: 390.53 355.40 349.17 339.47 322.78 319.23


9/13


9

INITIAL SHAPEX

Y

Z

L1 N1

L2

N5

N4

N3

N2

1000

500

375

6251000

1000

1000

1000

1000

X

Y

Z

L1N1

L2

N5

N4

N3

N2

1000

588.4

413.

497.71000

1000

1000

1000

1000

CASE II

X

Y

Z

L1

N1

L2

N5N4

N3

N2

500

375

625877.6

766.8

964.0

998.5

2300

CASE III

X

Y

Z

L1N1

L2

N5

N4

N3

N2

999.6

501.4

374.2624.4

975.0

973.6

1000.0

1000.0

1050.8

CASE IV

X

Y

Z

L1 N1

L2

N5

N4

N3

N2

880.0

511.8

375613.18

936.6

1059.3

1059.3

998.9

1066.1

CASE V

Figure 6. Optimum design obtained with MSC.Nastran program


10/13


10

PROBLEM II: CRITICAL LOAD MAXIMIZATION WITH MASS CONSTRAINT

In this case, the wing model is optimized for maximum critical load, considering the initial massof the structure (390 Kg) as a constraint. Table (5) presents the results obtained by

MSC.Nastran. The optimum shape is shown in Figs. (7).

Differently from problem I, table (5) shows that the most important design variables forbuckling load maximization are spar positions. This becomes evident when cases II and III are

compared. Although case III has more design variables (30), it delivers a poorer result (=0.40)

when compared to case II (=0.56) that has fewer design variables (27). The noticeable

difference is precisely the inclusion or exclusion of variables L1 and L2.

Table (5) indicates that the beam cross sectional areas and panel thicknesses do not play a role

as significant as in problem I. They do not vary dramatically from case to case. However, shapedesign variables are the driving force behind optimal designs for buckling load maximization.

The optimal skin panel thicknesses shown in table (5) are in accordance with the load

distribution presented in fig. (2). It can be observed that, from the clamped to the free end, thethickness decreases. This is expected since the compressive stresses in the upper skin panels arehigher in the neighborhood of the wing root. Rib and spar web thicknesses seem to benefit from

the inclusion of shape design variables in the optimization procedure as seen in case II and on.

In real applications an aircraft wing is never subjected to only one load case. Indeed, it is not

uncommon to have structures (wings, horizontal stabilizers, control surfaces) that experiencemany distinct load cases during operation, including but not limited to maneuver loads, gustloads, landing loads, etc. The load distribution depicted in fig. (2) and used in the present

simulations, reflect cruising aerodynamic loads. However, a more comprehensive optimizationshould consider the envelope of maximum loads rather than the loads associated with a

particular load case. Elegant techniques exist to handle buckling load maximization in a multipleload case situation (de Faria and Hansen, 2001).


11/13


11

Table 5. Optimal results for problem II

Variable Lower

bound

Initial

value

Upper

bound

CASE

I

CASE

II

CASE

III

CASE

IV

CASE

V

PLA11 0.3 0.80 2.00 0.70 0.46 0.48 0.50 0.48

PLA12 0.3 0.80 2.00 0.66 0.59 0.57 0.64 0.62

PLA13 0.3 0.80 2.00 0.66 0.73 0.69 0.73 0.72

PLA14 0.3 0.80 2.00 0.73 0.92 0.81 0.86 0.83PLA15 0.3 0.80 2.00 0.91 1.19 0.96 1.04 1.10

PLA16 0.3 0.80 2.00 1.02 1.45 1.06 1.18 1.25

PLA21 0.3 0.80 2.00 0.67 0.51 0.48 0.52 0.50

PLA22 0.3 0.80 2.00 0.63 0.67 0.57 0.64 0.67

PLA23 0.3 0.80 2.00 0.71 0.87 0.69 0.75 0.77

PLA24 0.3 0.80 2.00 0.99 1.18 0.84 0.87 0.96

PLA25 0.3 0.80 2.00 1.11 1.54 1.07 1.30 1.30

PLA26 0.3 0.80 2.00 1.36 1.91 1.45 1.48 1.55

PLA31 0.3 0.80 2.00 0.72 0.45 0.48 0.49 0.47

PLA32 0.3 0.80 2.00 0.69 0.55 0.58 0.61 0.61

PLA33 0.3 0.80 2.00 0.69 0.69 0.72 0.73 0.71

PLA34 0.3 0.80 2.00 0.69 0.82 0.84 0.86 0.82

PLA35 0.3 0.80 2.00 0.77 1.02 0.94 0.98 0.97

PLA36 0.3 0.80 2.00 0.86 1.28 1.08 1.11 1.10

RIB1 0.3 0.80 2.00 0.76 0.49 0.49 0.49 0.47

RIB2 0.3 0.80 2.00 0.71 0.52 0.49 0.52 0.49

RIB3 0.3 0.80 2.00 0.76 0.48 0.49 0.49 0.47

SPAR1 0.3 0.80 2.00 0.77 0.41 0.49 0.50 0.49

SPAR2 0.3 0.80 2.00 0.69 0.43 0.49 0.56 0.56

SPAR3 0.3 0.80 2.00 0.59 0.37 0.49 0.56 0.55

SPAR4 0.3 0.80 2.00 0.74 0.37 0.49 0.50 0.47

BEAM30 120.0 1256.1 1500.0 ----- ----- ----- ----- 1387.0

BEAM31 120.0 1256.1 1500.0 ---- ---- ---- ---- 1230.6

BEAM32 120.0 1256.1 1500.0 ---- ---- ---- ---- 1362.3

BEAM33 120.0 1256.1 1500.0 ---- ---- ---- ---- 1500.0BEAM34 120.0 1256.1 1500.0 ---- ---- ---- ---- 1356.0

BEAM35 120.0 1256.1 1500.0 ---- ---- ---- ---- 1000.0

BEAM36 120.0 1256.1 1500.0 ----- ----- ----- ----- 1297.4

BEAM37 120.0 1256.1 1500.0 ----- ----- ----- ----- 1191.7

BEAM38 120.0 1256.1 1500.0 ----- ----- ----- ----- 1191.9

BEAM39 120.0 1256.1 1500.0 ----- ----- ----- ----- 1191.7

BEAM40 120.0 1256.1 1500.0 ----- ----- ----- ----- 1191.7

N1 -1000.0 0.0 +1300.0 ---- ---- -496.10 -398.00 -368.70

N2 -1300.0 0.0 +1300.0 ---- ---- -496.10 -396.30 -368.87

N3 -1300.0 0.0 +1300.0 ----- ---- -496.10 -396.50 -273.44

N4 -1300.0 0.0 +1300.0 ----- ---- -494.14 +5.80 +45.43

N5 -1300.0 0.0 +1300.0 ----- ---- -495.61 -283.60 -350.65

L1 -400.0 100.0 +400.0 ----- 78.76 --- +95.0 +97.70

L2 -100.0 100.0 +100.0 ----- -187.90 --- -80.0 -80.80

Parameter Initial : 0.11 0.39 0.56 0.40 0.78 0.89


12/13


12

X

Y

Z

L1 N1

L2

N5

N4

N3

N2

1000

500

375

6251000

1000

1000

1000

1000

INITIAL SHAPE

X

Y

Z

L1N1

L2

N5

N4

N3

N2

1000

578.8

562.9358.41000

1000

1000

1000

1000

CASE II

X

Y

Z

L1N1

L2

N5

N4

N3

N2

1000.0

500

3756251495.6

999.5

1002.0

1000.0

503.9

CASE III

X

Y

Z

L1N1

L2

N5

N4

N3

N2

1001.7

595

4504501283.6

710.6

1402.3

999.8

602.0

CASE IV

X

Y

Z

L1 N1L2

N5

N4

N3

N2

999.8

597.7

455.8446.51350.7

603.9

1318.9

1095.4

631.3

CASE V

Figure 7. Optimum design obtained with MSC.Nastran program

CONCLUSIONS

Two optimization problems were investigated in this paper. It is noticed that even though the

basic geometry and the same sets of design variables are used, both problems have radical


13/13


differences with respect to design sensitivity. Problem I possesses high sensitivity with respect

to sizing design variables whereas in problem II the highest sensitivity is observed with respectto shape design variables. Generally, it is not possible to determine beforehand which subset ofdesign variables will be dominant in terms of sensitivity. Nevertheless, a preliminary sensitivity

analysis could certainly be helpful to identify relevant subsets.

In problem I, case III, the tendency of ribs to coalesce was detected. This kind of situation suitsbetter in topology optimization algorithms, a feature not yet available in MSC.Nastran, SOL200.With the resources currently implemented the best procedure would be to start off with a large

number of ribs and try to identify coalescence tendencies. Subsequently, a finer optimizationwould be conducted with the right number of ribs.

The only structural constraint involved in the previous optimization calculations was elasticbuckling. Nonetheless, other constraints are equally important, including but not limited to,

allowable stress/strain, fatigue, crippling and rivet bearing. The best finite element modelgenerated for optimization purposes will not be useful unless all appropriate constraints are

taken into account. If they are not, the resulting optimal design will be vulnerable to precisely

the constraint left aside. A simplified model such as the one studied herein provides an initialdesign but more realistic optimizations must be performed, perhaps in the component level,

before these structures become operational.

ACKNOWLEDGEMENTS

The authors wish to acknowledge FAPESP (Fundao de Amparo Pesquisa do Estado de So

Paulo) for the financial support provided to this research.

REFERENCES

1. Grihon, S. and Mah, M., Structural and multidisciplinary optimization Applied to AircarfDesign, 3rd World Congress of Structural and Multidisciplinary Optimization, Buffalo, NY,May 17-21, 1999.

2. Garcelon, J.H. and Balabanov, V., Integrating VisualDOC and GENESIS for

Multidisciplinary Optimization of a Transport Aircraft Wing, 3 rd World Congress ofStructural and Multidisciplinary Optimization, Buffalo, NY, May 17-21, 1999.

3. Garcelon, J.H., Balabanov, V. and Sobieski, J., 1999, Multidisciplinary Optimization of aTransport Aircarft Wing using VisualDOC, Optimization in Industry-II, Banff, Alberta,Canada, June 6-11.

4. Vanderplaats, G.N., Numerical Optimization Techniques for Engineering Design, 3rdedition, McGraw-Hill Book Company, 1999.

5. Powel M. J. D., Algorithms for Nonlinear Constraints that use Lagrangian Functions,Math, 1978.6. Moore, G.J., MSC.Nastran Design Sensitivity and Optimization, Users Guide, Version

68, The Macneal-Schwenler Corporation, 1994.7. de Faria, A.R. & Hansen, J.S., On Buckling Optimization under Uncertain Loading

Combinations, Structural and Multidisciplinary Optimization Journal, Vol. 21, No. 4,2001, pp. 272-282.

Documents

Optimization of Aircraft wing