Decomposition of a large optimization problem into several smaller subproblems has been proposed as an approach to making large-scale optimization problems tractable. To date, the characteristics of this approach have been tested on problems of limited complexity. The objective of the effort is to demonstrate the application of this multilevel optimization method on a large-scale design study using analytical models comparable to those currently being used in the aircraft industry. The purpose of the design study which is underway to provide this demonstration is to generate a wing design for a transport aircraft which will perform a specified mission with minimum block fuel. A definition of the problem; a discussion of the multilevel composition which is used for an aircraft wing; descriptions of analysis and optimization procedures used at each level; and numerical results obtained to date are included. Computational times required to perform various steps in the process are also given. Finally, a summary of the current status and plans for continuation of this development effort are given

Giles, G. L.

Wrenn, G. A.

English

NASA Technical Reports Server

N87-11746
MULTIDISCIPLINARY OPTIMIZATION APPLIED TO A TRANSPORT AIRCRAFT
Gary L. Giles
NASA Langley Research Center
Hampton, Virginia
and
G. A. Wrenn
Kentron International, Inc.
Hampton, Virginia
PRECEDING PAGE Bt.AHK NOT FILMED
439
https://ntrs.nasa.gov/search.jsp?R=19870002313 2020-03-23T14:50:22+00:00Z
TINTRODUCTION
Decomposition of a large optimization problem into several smaller subproblems
has been proposed as an approach to making large-scale optimization problems
tractable. To date, the characteristics of this approach have been tested on
problems of limited complexity (e.g., reference i). The objective of the
effort described in this paper is to demonstrate the application of this
multilevel optimization method on a large-scale design study using analytical
models comparable to those currently being used in the aircraft industry. The
purpose of the design study which is underway to provide this demonstration is
to generate a wing design for a transport aircraft which will perform a
specified mission with minimum block fuel. This paper includes (i) a
definition of the problem, (2) a discussion of the multilevel decomposition
which is used for an aircraft wing, (3) descriptions of analysis and
optimization procedures used at each level, and (4) numerical results obtained
to date. Computational times required to perform various steps in the process
are also given. Finally, a summary of the current status and plans for
continuation of this development effort are given (fig. I).
OBJECTIVE: TO DEMONSTRATE THE APPLI CATION OF MULTILEVEL
OPTIMIZATION METHOD iN A LARGE SCALE DESIGN
STUDY.
APPLICATION: TO GENERATE A WING DESIGN FOR A TRANSPORT AIRCRAFT
TO PERFORM A SPECIFIED MISSION WITH MINIMUM BLOCK
FUEL.
PRESENTATION
OUTLINE: • PROBLEM DEFINITION
• MULTILEVEL DECOMPOSITION
• ANALYSIS AND OPTIMIZATION PROCEDURES
• NUMERICAL RESULTS
• OBSERVATIONS
Figure 1
440
MULTILEVEL OPTIMIZATION APPLICATION
The multilevel optimization procedure is being applied to an L-1011 derivative
transport aircraft which is being studied by the Lockheed-California Company
as discussed in reference 2. The focus of this particular study is to design
a new wing to give minimum fuel consumption for a specified flight profile.
Design variables include overall wing geometric shape defined by aspect ratio,
sweep, total area, taper ratio and thickness ratio. In addition, variables
describing the wing structure within that shape are determined down to the
level of cross-sectional dimensions of stiffened-skin wing cover panels. As
overall wing geometry changes are made, the structure is reoptimized and the
static aeroelastic effects on aerodynamics are calculated but no aerodynamic
optimization of wing airfoil shape is performed. (See fig. 2.)
NEW
Wl Ni
(L-1011 DERIVATIVE)
OBJECTIVE
MINIMUM OF
BLOCKFUEL
CONSUMPTION
SMALL
TAI L
STRETCHED
BODY_
S sq. m, /R, hlc
NEWFLIGHT
STATION
CONSTRA INTS ON
PERFORMANCE
AERODYNAMI CS
STRUCTURE
NEW
PROPULSION
SYSTEM
Figure 2
DESIGN VARIABLES:
h/c, A, S, /R, Ct/Cr
AND
STRUCTURALCROSS SECTIONS
441
COOPERATIVE VENTURE WITH LOCKHEED
The study of the transport aircraft wing is being performed as a joint venture
with the Lockheed-California Company. Lockheed is using their integrated
structural design system which computerizes their conventional design methods
to perform such studies (reference 2). Parametric studies are used to
calibrate weight equations to perform overall configuration trade studies.
Structural sizing for this calibration is based on fully stressed design with
stiffened wing cover panels selected from design charts representing
predesigned cross sections. Aeroelastic considerations such as flutter and
gust are included in the Lockheed procedures. Multilevel optimization is
being applied at NASA Langley, initially to get the procedure implemented at
all levels for strength design and subsequently to include aeroelastic
considerations. Lockheed is under contract to provide sufficient design data
from their studies to allow NASA personnel to study the same configuration at
the same level of detail, (See fig. 3.)
LOCKHEED - CALIF.
CONVENTIONAL PARAMETRI C
STUDIES
• STRUCTURES:
INTEGRATED ANALYS I S
FULLY STRESSED DESIGN
PREDESIGNED CROSS SECTIONS
• AEROELASTIC CONSIDERATIONS
Figure 3
NASA LANGLEY
SYSTEMATI C MATHEMATI CAL
OPTIMIZATION
• MULTILEVEL APPROACH FROM
CONFIGURATION LEVEL DOWN
TO STRUCTURAL SIZING
LEVEL
• DI SCIPLINARY ANALYSES
COMPARABLE TO LOCKHEED'S
AS TO THEIR DEGREE OF
DETAI L
442
FINITE-ELEMENT STRUCTURAL MODEL
The finite-element representation of the structure was developed by Lockheed
personnel for analysis by the NASTRAN program used in their PADS system
(reference 3). Since the focus was on wing design, a fairly detailed model is
used for the wing structure and the regions of the fuselage necessary to get
proper representation of the wing-body intersection. The wing and wing-body
intersection structure is modeled primarily with rod and membrane panel
elements. The remainder of the structure (fore and aft fuselage, empennage,
engine, and landing gear) is modeled using beam elements. This NASTRAN model
was converted to be compatible with the Engineering Analysis Language (EAL)
system (reference 4) for analysis at NASA Langley. The resulting model has
641 joints for a symmetric half model. During design studies, only the cover
panels in the upper and lower surfaces of the main wing box (216 elements)
were resized (fig. 4).
641 JOINTS
1539 ELEMENTS (216 RESIZED)
Figure 4
443
THREE-LEVEL DECOMPOSITION
The decomposition for wing design is a particular case of the general
multilevel decomposition methodology described in reference 5. The wing
design process is decomposed into three separate optimization problems, as
shown in figure 5. At the top level, design variables such as wing
structural weight, aspect ratio, and sweep are used to minimize fuel
consumption subject to performance constraints. The optimum values of these
variables are then passed to the middle level as fixed parameters where the
distribution of wing box cover skin material is determined which will give a
minimum measure of constraint violation. Next, these optimum distributions
are passed to the bottom level where the optimum cross-sectional dimensions of
each of the stiffened panels are calculated. The optimization procedures at
the middle and bottom levels are used to minimize a single cumulative
constraint violation associated with that level. This cumulative constraint
is a differentiable envelope function of all individual constraints. The
particular envelope function used is the Kresselmeir-Steinhauser function
(reference 6). The cumulative constraints and their derivatives are passed
upward between levels. Iteration between the three levels is performed until
all constraints are satisfied. Analysis and optimization procedures used at
each level are discussed next.
t1, t2 ...
AIRCRAFT ]PERFORMANCE
A.... Qm, ,3 _mlc _ W
s
COVER SKIN
I STIFFENED II STIFFENEDPAN L 1 PAN L 2
MINIMIZE FUEL CONSUMPTION
SUBJECT TO PERFORMANCE
CONSTRAINTS
MINIMIZE CONSTRAINT VIOLATION
FOR WING STRUCTURE
MINIMIZE CONSTRAINT VIOLATION
FOR EACH PANEL
- CUMULATIVE CONSTRAINT,
KRESSELMEIR-STEINHAUSER FUNCTION USED
Figure 5
444
TOP LEVEL PROCEDURES
The Flight Optimization System (FLOPS) (reference 7) is used to perform overall
optimization at the top level. The objective is to determine the aircraft
wing configuration which minimizes block fuel consumption for a specified
mission. Performance constraints include limits on approach speed, field
length, and climb gradient thrust. Cumulative constraints from the lower
levels must also be satisfied. The standard version of FLOPS uses statistical
equations to calculate wing weight as a function of wing geometry. Modifi-
cations have been made so that the program can be implemented in the
multilevel optimization procedure by including wing structural weight as a
design variable and adding the cumulative constraint from the lower levels.
(See fig. 6.)
• FLOPS MISSION PERFORMANCE PROGRAM USED (REF 7)
• MODIFICATIONS TO FLOPS NECESSARY FOR MULTILEVEL IMPLEMENTATION
• WEIGHT OF WING STRUCTURE INCLUDED AS A DESIGN VARIABLE
• ADD A CONSTRAINT: CUMULATIVE CONSTRAINT FROM BoI-rOM
AND MIDDLE LEVELS
Figure 6
445
MIDDLELEVELPROCEDURES
The optimum distribution of material in the wing box cover skins is calculated
by the middle-level procedures. Displacements and stresses are calculated
using the model in figure 4 as input to the EALsystem. Analytical
derivatives of these quantities are calculated using the procedures described
in reference 8 which are implemented as sequences of input statements to EAL.
The design variables used in optimization are coefficients in a polynomial
expression for the cover thickness distribution. The distribution currently
being used is illustrated in figure 7. As indicated on figure 5, the
objective function is a cumulative constraint from the middle and bottom
levels with a fixed weight of the wing box covers from the top level specified
as a constraint. Optimization is performed using CONMIN(reference 9) in a
sequence of steps in which the results from the structural analysis are
approximated by linear extrapolation.
• EAL USED FOR STATIC ANALYSIS AND DERIVATIVES
ANALYTICAL DERIVATIVES USED FOR THICKNESS VARIABLES
• DESIGN VARIABLES COEFFICIENTS IN EXPRESSION FOR COVER
THICKNESS DISTRIBUTION
TSKIN = C0 + C1 II-_) + C2 11-13) 2
• OBJECTIVE FUNCTION MINIMUM CUMULATIVE CONSTRAINT FROM:
MIDDLE LEVEL - WING TIP DISPLACEMENT
BOFrOM LEVEL - PANEL CUMULATIVE CONSTRAINTS
• CONSTRAINT = FIXED WEIGHT OF WING BOX COVERS
• PIECEWISE LINEAR OPTIMIZATION PROCEDUREUSING CONMIN
Figure 7
446
INITIAL SKIN PANEL DESIGN VARIABLE LINKING
The thickness properties of the finite elements representing the wing box
cover skins are described in the spanwise direction by the quadratic
expression in terms of the nondimensional parameter "B" (B=O at the wing root
and B=I at the tip) shown on figure 8. Two quadratic segments are used, one
inboard of the engine pylon and the other outboard. A constant thickness is
specified in the chordwise direction. The upper and lower wing box cover skin
properties are taken to be symmetric with respect to the wing middle surface.
The six coefficients of the two quadratic expressions are the design variables
used during optimization. This linking scheme is used to reduce the number of
design variables during initial testing of the multilevel optimization
procedure. It is recognized that this simplified linking restricts the
possible distributions available for optimization and these restrictions will
have to be removed after the initial testing phase.
TSKIN : CO + C1 (1 - 13)+ C2 (1 - 13)2-_ _____....j_ _
\
_CONSTANT
/ • sYMMErR,cUPPER
/ ANDLOWERWiNG
"-._/ BOX COVER SKINS
Figure 8
447
BOTTOM LEVEL PROCEDURES
Each of the 216 wing box cover panels is optimized by the bottom level
procedures. Although properties of corresponding panels on the upper and
lower surfaces are taken to be the same, the panel loads are not the same.
Therefore, each pair of panels is optimized in order to assure consideration of
the panels with the critical loadings. The design variables are the cross-
sectional dimensions of a stiffened panel, as shown in figure 9. The
objective function is a cumulative constraint composed of contributions from
five stress constraints and eight buckling constraints that are considered.
The CONMIN program is used for optimization. After each panel is optimized,
an optimum sensitivity analysis is performed to get derivatives of the
cumulative constraint with respect to parameters such as panel length, width
and stress resultants which are passed down from the middle level. The
algorithm described in reference i0 is used for these calculations. Finally,
these optimum sensitivity derivatives are combined wih structural response
derivatives from the middle level to form cumulative constraint derivatives
which are subsequently used in the middle-level optimization process.
PANEL OPTIMIZATION OPTIMUM SENSITIVITY ANALYSIS
• DESIGN VARIABLES • PARAMETERS
T
_-bf-_ tf Nx
• FAILURE MODES aO 6g_
STRESS (5) c_N ' 6 N
BUCKLING (8) x y
• CONMIN USED CUMULATIVE CONSTRAINT CALCULATION
X
-- _!. • • •
c_tI c_Nx 0t I
Figure 9
448
MULTILEVEL OPTIMIZATION IMPLEMENTATION FOR L-1011 DERIVATIVE WING
The general characteristics of the computer programs used in each of the three
levels have been described in the previous discussions of the levels. The
original intent was to transfer all data between levels via the Relational
Information Management (RIM) system (reference 11). Since theprocedures in
the middle level are all related to the EAL structural analysis system, its
data base was used for all data communication within the middle level. It was
found that the bottom level was tightly coupled to the middle level in terms
of types and quantities of data that had to be shared. Consequently, the
bottom level was implemented as an EAL processor and utilities described in
reference 12 were used to provide data communication to the EAL data base.
The relatively small amount of data communication required between the mid-
dle level and top level is handled using the RIM system. (See fig. I0.)
LEVEL
=-_
TOP MISSION
PERFORMANCE
LOADS CALCULATION I
STRUCTURAL ANALYSIS AND I
RESPONSE DERIVATIVES I
STRUCTURAL SIZING I
STIFFENED PANEL
OPTIMIZATION
MIDDLE
BOTTOM
Figure 10
m
E RI
A
L I ]
D MI
A
T
A
B
A
JE
449
MINIMUM WEIGHT SIZING BY INDIRECT METHOD
To assess the results being produced by the bottom two levels, an indirect
method of calculating a minimum weight design was employed. The middle and
bottom levels were used to calculate minimum values of the middle-level
cumulative constraint for four values of wing box cover weight. These
optimized designs are indicated by the circular symbols on figure II. The
point above the horizontal axis is infeasible since the cumulative constraint
has a positive value and the three points below the axis satisfy all
constraints but are overdesigned. The minimum weight design is located where
a line through these points intersects the horizontal axis as shown on the
figure. This design is heavier than the minimum weight design produced in the
Lockheed studies. The difference is attributed to the restrictions imposed by
the initial design variable linking scheme, figure 8, that is being used for
testing purposes.
Q>O,
V l OLATED
i
.2 --
.1 --
--° 1 m
--.2 B
--, 3 b
CUMULATI VE
CONSTRAINT 0
Q
Q< O,
SATISFIED
I
8
\
INFEASIBLE C_
k/MINIMUM WEIGHT
/ 10 x,_ 14000 WING BOX
t- LOCKHEED _ COVER WEIGHT,
DESIGN
3 829 UPPER
4 847 LOWER
\
FEASIBLE,--- "C_
OVERDESIGNED
Ib
Figure 11
450
TYPICAL SUMMARY OF COMPUTATIONAL ACTIVITY
A summary of the computational activity for the major tasks involved in the
operation of the middle and bottom levels is shown in figure 12. Both
normalized CPU time and I/0 count are given. Performing a static structural
analysis and calculating derivatives of the response quantities involve
considerable computational activity. A large portion of CPU time is required
for panel optimization at the bottom level where 216 separate optimization
runs are made. Only a small amount of I/0 activity is required in these
calculations. The CPU time required for optimization at the middle level is
an order of magnitude less than that required for the structural analysis and
derivatives that are used for linear approximation during optimization. Total
CPU time, I/0 count, and cost are shown at the bottom of the figure for five
piecewise linear optimization cycles on a CDC Cyber 175 computer.
NORMALIZED NORMALIZED
TASK CPU TIME I/O COUNT
INITIALIZE
STATI C STRUCTURAL
ANALYSIS
STATIC DERIVATIVES
PANEL OPTIMIZATION &
SENSITI VITY ANALYSI S
(= 35 ITERATIONS/PANEL)
LINEAR OPTIMIZATION
CYCLE FOR
(10 ITERATIONS)
•O52
•174
• 137
• 613
•024
.134
• 277
•243
•025
•321
1.000 1.000
-'--216 PANELS
FOR 5 PIECEWISE
LINEAR CYCLES
ON CYBER 175
CPU TIME 650 sec
I/O COUNT 34O0O
COST $350
Figure 12
451
CONCLUDING REMARKS
The current status of the implementation of the multilevel optimization
procedure is summarized on figure 13. The aircraft wing design process has
been decomposed into three levels. The bottom two levels have been
implemented using the EAL system and have been successfully tested. This
initial testing resulted in the demonstration of an indirect method for
minimum weight design which may prove to be an attractive alternative to
conventional methods that have been used in the past. The three-level system
can be tested when the FLOPS program is incorporated at the top level and
efforts to demonstrate the application of the multilevel optimization method
on a large-scale design study are continuing.
• AIRCRAFT WING DESIGN DECOMPOSED INTO THREE LEVELS
• INTEGRATION AND TESTING OF BOTTOM TWO LEVELS SUCCESSFULLY COMPLETED
• MINIMUM WEIGHT SIZING BY INDIRECT METHOD DEMONSTRATED
• FLOPS PROGRAM TO BE INCORPORATED AT TOP LEVEL
• STUDY CONTINUING
Figure 13
452
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Multidisciplinary optimization applied to a transport aircraft

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