Simultaneous Aerodynamic and Structural Design Optimization (SASDO) for a 3-D Wing

16 Pages·2001·0.89 MB·English

by  Gumbert, Clyde R.

AIAA 2001-2527
Simultaneous Aerodynamic and Structural
Design Optimization (SASDO) for a 3-D Wing
Clyde R. Gumbert
NASA Langley Research Center
Hampton, Virginia
Gene J.-W. Hou
Old Dominion University
Norfolk, Virginia
Perry A. Newman
NASA Langley Research Center
Hampton, Virginia
preprint of the paper presented at the
AIAA 15 th Computational Fluid Dynamics Conference
June 11-14, 2001
Anaheim, CA

AIAA 200"1-2527
Simultaneous Aerodynamic and Structural
Design Optimization (SASDO) for a 3-D Wing
Clyde R. Guml)ert*
NASA Langley Re.search ('_nler, Hampton, I:4 23681
Gene J. -W. Hout
Old Dominion Univer._'ily, NolJ'olk, 1:4 23529-02_7
Perry A. Newman ¢
XASA Langley Research (:_:nh:r, Hamplon. I:4 23681
Tile formulation and implementation of an optimization method called Simultaneous
Aerodynamic and Structural Design Optimization (SASDO) is shown as an extension of
the Simultaneous Aerodynamic Analysis and Design Optimization (SAADO) tnethod. It
is extended by the inclusion of structure element sizing parameters as design variables and
Finite Element Method (FEM) analysis responses as constraints. The method aims to
reduce the computational expense incurred in performing shape and sizing optimization
using state-of-the-art Computational Fluid Dynamics (CFD) flow analysis, FEM struc-
tural analysis and sensitivity analysis tools. SASDO is applied to a simple, isolated, 3-D
wing in inviscid flow. Results show that the method finds the same local optimum as
a conventional optimization method with sonic reduction in the computational cost and
without significant modifications to the analysis tools.
Nomenclature _Q_ change in flow solver field variables due to
b wing semispan better analysis convergence
('D drag coefficient AQ2 change in flow solver field variables due to
CL lift coefficient design changes
pitching lnonlent coefficient H aerodynamic state equation residuals at,
each CF1) mesh point
wing root chord
Cr
wing tip chord IH/R_)I norm of the residual ratio, current/initial
Ct
F design objective functioll s surface area
g design constraints S semispan wing planform area
K stiffness lnatrix u structural deflections (state variables)
L aerodynamic loads ,_._?11 change in deflections due to better analysis
M_. free-stream Math number convergence
7_ unit normal vector Au2 change in deflections due trOdesign changes
P local aerodynamic pressure tI' wing weight
P compliance, the work done by the aerody- X CFD volume mesh coordinates
namic load to deflect the structure
location of wing root, leading edge
.I_LE
free-streana dynamic pressure __
q,>,_ chordwise location normalized by local wing
Q flow-field variables (state variables) at, each c
section chord
(_FD mesh point
longitudinal location of wing tip trailing
a?t
*Research Engineer. Multidisciplinary Optimization Branch. edge
M/S 159, c.r.gumbert _larc.nasa.g ov root section lnaxinllllll ca.tuber
tprofessor, Department of Mechanical Engineering, AIAA 2r
member, ghow_lions.odu.edu o free-stream angle-of-attack
_Senior Research Scientist. Multidisciplinary Optinfization i:_ design variables
Branch, M/S 159, p.a.newman__larc.rla.,_a.gov
structural element size factor
Copyright @ 2(101 by the Am,_rican lnstitut_ _of Aeronaulics and F
Astronautics, Inc No copyright is ass_rted in the United States line search parameter
under Title 17, US. Codt, The U.S Government has a royalty- "_
tree license to exercise: all righ'_s under the copyright claimed hereto operator which indicates a.change in a vari-
for Governmental Purl-,OSeS All other rights are reserved by tIw
able
copyright own_,r
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AMERICAN INSTITUTE OF AEfCONAIrT1CS AND ASTRONAUTICS PAPEIq 2001 2527
,£,Sf convergence tolerances for function and gra- opiimization 2 discusses both NAND and SAND pro-
dients
cedures in the context of current steady aerodynamic
A adjoim variabh' for Q opt imization research.
P adjomt variable for u For single-discipline design problems, the distinc-
T twist angle al wing tip, positive for leading tion between NAND and SAND procedures is fairly
edge up clear and readily seen. With respect to discipline fea-
sibility (i.e., convergence of the generally nonlinear,
Subscripts iteratively solved state equations), lhese procedures
d deflected shape can be viewed as accomplishing design by using only
j ,jig (undefleeted) shape very well converged discipline solutions (NAND), or
as converging a sequence of discipline solutions from
s wing surface mesh
poorly to well as the design progresses (SAND). How-
v wing volume mesh
ever, the problem formulation and solution algorithms
may differ considerably. About twenty SAND refer-
Superscripts
ences are quoted by Newman el al. 1 and Newman et
* designates updated value
al.;" these references discuss a variely of formulations.
' gradient with respect t.o design racial)los algorithms, and results for single-discipline problems
(mostly (;FD applications) in the seuse of SAND as de-
Introduction
fined above. For mull!disciplinary design oplhnization
Simultaneous Aerodynalnic Analysis and Design problems, the distinction between NAND and SAND
OptimizAtion (SAADO) is a procedure that incorpo- is somewhat, blurred because !here are feasibility con-
rates design improvement within the iteratively solved siderations with respect to all the individual discipline
(nonlinear) aerodynamic analysis so as to Achieve flflly state equations, as well AS with respecl to the nmlti-
converged flow solutions only near an optimal design. disciplinary syslelll compatil)ility and constraints. A
When SAADO is applied to a flexible wing rather number of the papers in Her. 3 discuss MDO fur-
than a rigid wing, the linear Finite Element Method mulations And algorithms that are called SAND-like:
(FEM) solution is iteratively coupled with tile nonliu- however, not all of these latter MDO procedures ap-
ear (?omputationa] Fluid I)ynamics (CFD) solution. pear t,o agree with the sense of SAND defined above
VVhen design variables that control structural element and used herein: one that does is Ref. 4.
size Are included, it is renamed Simultaneous Aerody- The computational feasibility of SAADO for quasi
namic and Structural Design ()pt, imization (SASDO). I-D nozzle shape design based on the Euler equation
Overall computational etficiency is achieved because (T1) approximation was demonstrated by Hou el. al. s
the many expensiw' iterative (nonlinear) solutions for and Manif Application of SAADO for turbulent tran-
non-opt imal design parameters are not converged (i.e., souic airfoil shape design based on a 2-D thin-layer
ol)tained) at each optimization step. One can ob- Navier- Stokes (:F l) approximation was demons! rated
lain the design ill the equivalent of a few (rather and reported in a later paper by tlou et at. r Both of
than many) multiples of the computational tittle for these application results are summarized and briefly
a single, fully converged coupled aero-structural anAl- discussed in Bet. 1. The al)plication of SAADO for
ysis. SAADO and similar procedures for simultaneous rigid 3-I) wing design bAsed on the Euler CFD ap-
AnAlysis and design (SAND) developed by others are proximation WAS presented ill Ref. 8. These SAADO
noted and discussed by Newman et al. l These SAND procedures utilized quasi-anAlytical sensitivity deriva-
procedures appear best suited for al)plications where tives ol)tained from hand-differemiated code for the
the discipline analyses involved ill the design are non- initial quasi I-D application s'; and from automatically
linear and solved iteratively. Generally, convergence differentiated code for both lhe :2-1) airfoil appliea-
of these discipline analyses (i.e., state equations) is !.ion r and the 3-D rigid wing Application. s Different
viewed as An equality constrainl in an optimization opt.infization techniques have also been used in these
tm)blem. From this latter point of view. the SASDO SAADO procedures.
method proceeds through infeasible regions of tile de- Tile extension to multidisciplinary analysis with
sign space which inclu(tes not only the design variables shape design variables only was presented in Ref. 9.
3, bu! also the state variables Q and u. A further ad- Our initial resuhs from SASDO are given in this pa-
vantage of SASI)O is the efficient utilization ofexist.ing per. The analysis i)roblem, the objective function, and
discipline analysis codes (without internal changes), the constraints are the same as those used in Ref. 9.
augnlel|led with sensitivity or gradient informatioll, Tha! is, changes in design variables are sought to
an(l yet effectively COUl)led more tightly than is done in produce improvement in the lift-to-drag ratio of a sin>
conventional gradient-based optimization procedures, pie wing subject t,o both aerodynamic and structural
referred to as nested analysis and design (NAND) solution-dependent constrainl.s. These constraints are
procedures, l A recent overview of aerodynanfic shape tile difference between tile lift and weight, the pitch-
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AMEI(1CAN INSTITUTE OF AEI_ONAUTICS AND ASTIRONAUTICS PAPER 2001 2527
ing nloment coefficient, and the compliance, a function
representing work done 1)3' the aerodynamic load to
deflect the structure. There are also geometric (:Oll-
straints. ( )
The flexible wing studied here is fornmlated as a
static aeroelast.ic problem. Similar problems have
been used as examples in Refs. 10-15 to study var-
ious solution strategies tbr multidisciplinary analysis ¢>
and optimization. In particular, Arian 12 analyzed the
Hessian matrix of the system equations to derive the
mathematical conditions under which the aeroelastic
ol_timization problem can be solved in a "loosely'" con-
pied manner. The nmltidisciplinary research of Walsh
et al. 13' 14emphasized engineering aspects of integrat-
ing high fidelity disciplinary analysis software and dis-
tributed computing over a network of heterogeneous
computers. The aeroelastic analysis results of Beuther
et al. 15 were verified with experimental data.
Only a limited amount of literature related to aero-
elastic problems has elaborated on the coupled sen-
sitivity analysis. Kapania, Eldred and Bartheleno': t(;
Arslan and Carlson: 17and Giunt a and Sobieszczanski- _-Xz? _tr -_ xt "-
SobieskilS derived global sensitivity equations (GSEs);
Fig. 1 Description of semispan wing pm'mneterl-
some matrix coefficients in these GSEs were evalnated
zation.
by finite differencing. Guinl.a 1!'later introduced modal
coordinates to approximal.e the elastic displacement Fig. 1. The baseline wing section varied linearly from
vector in order to reduce the size of the GSE. Newman, an NA(!A 0012 at. the root to an NACA 0008 at the tip.
Whitfield, and Anderson _° used the complex variable The wing structure consists of a skin, ribs, and spars.
approach to obtain the aeroelastic sensitivily deriva- The ribs and spars consist of shear webs and trusses.
t.ives, whereas [_euther et al. 15 employed the adjoint Six spanwise zones of the structural model are defined
variable approach to derive the aeroelastie sensitivity as depicted in Fig. 1. The relative sizes of the skin and
equations. A niat.helnaticalstudy of the coupled non- web thicknesses and the truss ('ross section areas are,
linear, incompressible aeroelastic analysis and sensitiv- fixed within each zone. Each zone is assigned a param-
ity analysis problems has been given 1)3' Gha.ttas and eter F which multiplies all the thicknesses and areas
Li. =q Hecent results on aeroelastie sensitivity analysis of the structural elemenls in that zone. The specific
and optimization can be found in Refs. 22-24. Partic- parameters selected as design variables in the sample
ularly, Maute et al. ''a and Hou and Satyanarayana '-'4 optimizat.ioll problems are identified in the section en-
explicitly formulated the deflection update and the titled R.esults. The objective function to be minimized
load transfer between the separate flow and structures was the negative of the lift-to-drag ratio. -L/D. Both
solvers as part of the coupled sensitivity equations. coupled solution-del)endent arm geometric constraints
In the present study, the coupled sensitivity equa- were imposed.
tions are constructed by differentiating the aeroelastic The solution-dependent constraints were
state equations and solving them by a Generalized
Gauss-Seidel (GGS) method. 1" The present SASDO • lower limit on the difference between the total lift
concept is very similar to that of Ghattas and others, and the structural weight, ('t * 5; * qx -- II
Befs. 4, 21,25, 26, but. differs in the implementation
details as described later. • upper limit on compliance, P = .}pu . rids
• upper limit on pitching moment, (',,,, in lieu of a
Problem Description
trim constraint
To evaluate the efficacy of the SASDO procedure for
a problem involving inultidisciplinary analysis, it. is al)- The purely geometric constraints were
plied herein to a simple, isolated, flexible wing. The
wing shape consisted of a trapezoidal planform with a • nlinilnunt leading edge radius, in lieu of a mann-
rounded tip. It was parameterized by fifteen variables: facturing requirement
five described the planform, and five each described
the root. and tip section shapes. A schematic of the • side constraints (bounds) on the act.ire design
wing and its associated shape parameters is shown in variables
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AMERICAN [NSTITUTE OF AEIqC)NAIJTICS AND ASTRONAUTICS PAPER 2(101 2.527
SASDO Procedure
and equality constraints
Formulation
OR/k() oR oxa_, -
The flexible SASDO approach formulates lhe desigu R(Q, Xd_(.d, u),.d) + oO "_ + __axu
(7)
Ol)t.mlizatiou problem as follows:
+\_ox_," j+_lAd=O
lUill l,'(O. X&,(3, it). u, 3) (1) and
d,(_ ,tZ
subject to i,(A; (_), _). - L(Q, .\_, (,_. ,,)) - _°I_x:)._
+ h-__ Au+ _u A,'_ (8)
:h(Q,X,_,(,Lu),u,d)_<O: i= 1,2 ..... m (2)
I°K u :_l. _Jhd, _ V_A4-- n
where the flow field Q and the structural deflection u
are solutions of the coupled flow equation Note that Eqs. (5) through (8) are linearized approxi-
mations of Eqs. (1) through (4), respectively.
R(Q, Xa,(d, u), d) = U (3) In this formulation, neither lhe residual of the non-
linear aerodynamic field equations, R(O,.\',d). nor
and the finite element structural equation
that of the st,ructures equation, hu- L, is required to
I)e zero (reach target) until the final optimum design is
K(X/(3), 3)u = L(Q, .\'d,_(3, u)) (4)
achieved. The linearized prol)lem of Eqs. (5) through
"l'l.' deflected volume mesh, Xd,, is determined by' (8) is difficult, to solve directly because of the number
t.hc deflected surface mesh, Xd.,, a.s .Ym = .Y,t, (-\'_t.,). of desigu variables and equality constraiut equations.
The deflected surface mesh is a resuh of the jig shape
Direct diffi r_nt*atton method
augmented by the elastic defl,wlion, u. as -Ya.. =
One way to overcome this difficulty is by the di-
X j(3) + u. The two disciplines are coupled through
lhe deflection, u, and the [<lad, L. rect difforentiation inethod. In this method AQ, Au,
and Eqs. (7) and (8) are removed altogether fronl
Recall that Q, R, and Xa,, are very large vectors.
the linearized probMlt by direct substitution. This
This h),'mulation treats the state variables, Q and u,
is achieved I)3,"expressing At"2 and ._Xu as functions of
as part of the se! of indel)endent design variables, and
..Xd.
considers the state equations to be constraints. Be-
AQ = AOl + AQ.eA3
cause satisfaction of the equality constraints, Eqs. (3)
Au = Aul + Au,,Ad (9)
and (4), is required only at the final optinmln solution,
the coupled steady-state aero-structural field equa- where vectors AQa and A-u1 are corrections in the
lions are not converged at every design-optinfizatioll aeroelastic solution due to the ilnprovemenl of cou-
iteralion. The easing of that restriction is expected l>led aeroelastie aualysis, while matrices AQ., an(l AWe
to significantly reduce tl,e excessively large compu- are corrections due to changes in the design variables.
t.ational cost incurred in the conventional approach. These vectors and mah'iees are solutions of the follow-
llowever, this advantage wouht likely be offset by the ing coupled sels of equations, obtaine(I from Eqs. (7)
very large increase in lhe Ilult|})el' of design variables and (8):
and equality constraint functions, unless some reme-
ill? A,,_ OH OXa,, *_
dial procedure is adopted. _7_Q*.-A_ I h' ox,_,, 0a,,
(io)
Approximations
Ix',._t! = 7OL70-A- Q1 + oxOL_, __XAa..t*l
The SASI)O method begins with a lilwarized design
optilnization problem solved for the rues! favorable
where, for the linear FEM, 1(. - L = 0 at, every iter-
change in the design variables, A J, as well as for the at|oiL and
changes in the state variables, ._XQ and ,.Xu: that is,
_AQ=, + ox_,, ox_, _- ,i + Au,_,) + _ = 0
lnin F(Q, Xa, (d, u), u, d)
£.'x,L ",Q.A,
+_AQ+ + _x.,-ox-.-,-)S,Au (5) h',.._ it'-,= =OL__Q.t')''a_2 + ObX_La,, vOXXaa,._ g_"\',.1+ ,.Xu=,) (11)
\
- (\e_.x_,-, \-a_ +_ .a ) u
subject to inequality (-onstraints Note that tb.e nulnl)er of columns of mat.rices AQ=,
and Au:e, is equal to the number of design variables.
:I,(Q..\a,_(/, .), .. 3) + _ + A. d. Thus the computational cosl of Eq. (1l) is directly
proportional to the number of design variables.
(6)
OQ "_ \ oX_a,,,. OOXXa," .I + O(cid:0) ) A new linearized l)roblem with A3 as the only design
i= 1.2 ..... m variables can t)e obtained t)3, sul)stituting Eq. (.q) into
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AMERICAN INSTITUTE OF AEFtONAUTICS AND ASTRONAUTICS PAPE,q 2001 2527
Eqs. (5) and (6) for AQ and Au. so as to rewrite the expression of the object ive function
in Eq. (5) in terms of 3,, p, and A3 as
Illi n F(Q, X, it,3) + O-_FQA,q,41 F(Q, Xdv(/_), 'it), ll_ J) + _T R + liT(Kit -- L)
a_
[_,r 4- OF OX_,,'_ +{( \_oOxF_ OXa,. _\',J + _)
+ _0_, -- OX.,, OX_.] Aul (12)
OF
+I_AQ 2+ OXOa,F. _OX¢_, "V-./'.
,or< u [oK oL ox,, -,_l,A,2,
+liT \ O_'3 + _(-_2 u OXd,. OXa, "Y'J ]J
+ ( 0O-2F7,.OoXxa,_,.., + Oe5F)7 ,_._,,,)- + _OF}/k_ ., (15)
Note that the terms in the brace represent the gra-
dient of the objective function. The terms 3`TR and
subject to
IIT(Ku -- L) indicate the effect on the design opti-
mization fornmlation due to errors in the aeroelastic
.qj(O, .X,u, d) + 0QAQ1 analysis. Furthermore, in the linear sense, the adjoint
variables and the solution errors can be related by the
following equa! ions:
+ JLq-_Ao, + __(\". Au2)
!. oO "_- ONe,, ox_s " a+
% OF HT (16)
_- o_,.Xu.2+ °o-,m'_-}Az_<0;- i 1.2 ..... 7n 03`
(1:3)
The appearance of AQ1 and Aua in the formulation aud
indicates the difference between the SASDO (SAND) OF (Ixu L)T (17)
Of
method and the conventional (NAND) aerodynamic
optiinization method. The AQ1 and Aul not only These equations have been mentioned, for example.
constitute a change in Q and u, but. also play all impor- by Pierce and Giles-"; and Venditti and Dannofal 2s
tant role in defining the objective function of Eq. (12) for aerodynamic problems.
aim the constraint violation of Eq. (13). We can di- In the typical optimization problem there are many
rectly solve Eqs. 10 for AQa and Aul, and, in fact, design variables. When one can also pose the op-
timization problem such that there are only a few
in previous SAADO applications that is how AQ1 was
determined. However, since AQt and Aul, as shown output quantities for objectives aim constraints or. in
in Eq. (10), represent a single Newton's iteration on the extrenle, combine the constraints and objective
the coupled flow aim structures equations, it is possi- function into a single cost fimctiou, the adjomt ap-
ble and less comput ationally expensive t.o approximate proach to sensitivity analysis has the advantage that
their influence on the solution Q and u by several New- the adjoint solutions are independent of the lmmber of
ton's iterations of the coupled aeroelastic equations. design variables. >" However, when the discil)lines are
That. is, AQ1 and Aui are not deternfined explicitly, loosely coupled, this approach is ilnpractical since the
but rather the first three terms of Eqs. 12 and 13 are coupled sensitivity analyses would require an adjoint
viewed as updated values of F and gi. Note that the for each disciplinary output being transferred, i.e., the
diseretized loads and deflections. In a tightly or iln-
terlns in parentheses in Eqs. (12) and (13) are ap-
proximated gradients of the objective and constraint plicitly ('oupled multidisciplinary analysis the adjoint
functions. Once established, this linearized problem approach may prove practical since this system is anal-
can be solved using any mathenlatical progratnming ogous to a single discipliue.
technique for design changes, ,-.M3. l'/esults presented Line Search
in this paper are computed using this direct differen-
A one-dimensional search on the step size parame-
tiatiol_ approach.
t.er 3' is then performed in order to film the updated
values of A/3, AX, AQ, and Au. Given the search
Adjoint method direction Ad det.ermiued hy either the direct, differen-
An alternative way to remove AQ and Au from the tiation naethod (Eqs. 12 and 13) or the adjoint method
equivalent, this line search functions to adjust its inag-
linearized problems, Eqs. (5) and (6), is the adjoint
variable method. The adjoint variables, 3,and p, can nitude so as to simultaneously ensure better results for
be introduced as the solutions of both design and analysis (converged solutions). The
step size parameter _, plays the role of a relaxation
factor in the st.andard Newton's iteration. The search
(aO-RO]T),. ""= (_O)Lrl' - (_O,F IT procedure employed solves a nonlinear optimization
problem of the form
K/t=( e,L e,xd,. ) r (14)
rain F(Q*, X*, u*, /3") (18)
oH ox,,.) r (0v a___._r OF
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AMERICAN INSTITUTE OF AEIRONAUTICS AND ASTRONAUTICS PAPER 2001-2527
r ................................
sul)jeet to Specify: Initiel!ze:
' Design Problem Geometry
Final Convergence F_I_I MC_c_edl]ions
,qi((-2.\-*, u., _'3") < 0; i = 1,2 ..... m (19)
J i Convergence Level
R(Q',.\", ;r) = 0 (20) |I ........... Pasrti.atL'ly-"t:.C.oAn.ve.rn_"Y..-..*.;; ,........ P._art,ie_lly.*yC.o.nAver.g,ed;_....,._.:
]i l-o o-cl! i I
and
I _ I: -" " .-, _,n,_,,, ,
h(.\", a")." = L(Q', X') (21) L]I_I ',I Geometry I ' Ana_iyeis ' i ',I Geometry F"I I
.__I ,nclMesl_I T ', I :I ,na_ l
["-; I _ : I 'IS..eiuvit_, L._ i
where step size ") is the only design variable. Again II '" ' I_'_:_.'1 ,:II,,.f;I 1"7 -"Struclurel I
it is noted for emphasis that the equality constraints.
:::::::::::::::::::::::::::::::::::::::::::: ....;:::::::::::
Eqs. (20) and (21), are not required to be zero (reach
target) until the final optinmm design; violations of I ! I Tighten J _ ' ' ' " I
, . . , ' reproved
I Convergen_ _ O_;; _ Optimizer _ Deeign
these equality constraints nlust simply be progressively : l U,ve, I I " I ' ': :
reduced until the SASDO procedure converges. Pertialmy Conver_ld Design
The updated Q* and u* can be viewed as Q* = Q+
&Q and u* = u + ,:_ku. where _Q and Xu satisfy the Fig. 2 Diagram of SASDO procedure.
first order approximations to Eqs. (20) and (21). That
Biros and Ghattas. :''_ In addition, lhe line search st.ep
is, XQ and -ku are the solutions of Eqs. (7) and (8)
(step 6) and the convergence tightening step (step 8)
where, in Eq. (9). ,_k3 is replaced by _,_" = h,__k3.
were not included in the Biros and (;hatta_s method.
('.onsequently. Q* = Q + AQ1 + ")-kQ._,.._ko' and u* =
A schematic of tile present SASDO procedure is shown
u+A uI+"I,-3, u._,A3 are readily available once ") is[bund.
in Fig. 2. The dashed box, labeled "Partially Con-
The A()l terms appearing in the above SASDO for-
verged System Analyis," depicts lhe coupled analysis
mulation are due to better ,'onvergence of the coupled
iteration loop. Steps 1 and 2 of' the pseudocode: that
analysis, whereas the _()2 terms at'<'due t.o changes in
labeled "'Partially (!onverged Sensitivity Analysis"
the design variables. 17]fact, ..XQ._,and Au._, approach
depicts the coupled derivatiw, iteration loop. St.ep 3:
the flow field and deflection sensitivities, Q' and uI, as
that labeled "Partially (:onverge(I l)esign" depicts the
the solution becomes better converged.
design steps, Steps 5 8 of the pseudocode. Specific
hnplementation computational tools and methods used to l)erform
'I'17(' following pseudocode shows algorithmically the tasks depicted by the solid boxes in Fig. 2 are
how the method was implemented. identified in the next section.
set initial analysis convergence tolerance, _-
set initial solution vectors, Q and u, Computational Tools and Models
set, initial design variables. :3 Major comlmtations in this SAS|)O procedure are
do until converged
performed using a collection of existing codes. These
1. solve coupled aeroelastic analysis Eqs. (3) & (4), codes are executed by a separate driver code and
partially converged to _"
scripts that implement the SASDO procedure as ,just.
2. compute F and g
discussed. Each code runs independently some simul-
3. solve coupled aeroela.stic sensitivity analysis.
taneously, on different processors, and the required
Eq. (I I), partially converged to _-"'
I/O transfers between l,hein, also directed by the
4. compute ._X3 terms of Eqs. (5) & 05)
driver, are accomplished via data files.
5. solve optimization problem Eqs. (5) &"(6) for ,.X3
The aerodynamic flow analysis code used for this
6. solve Eqs. (18) through (21) for line search pa- study is a version of the (:FL3D code. a" Only Eu-
rameter, ter analyses are performed for this work, although the
7. update d, u. and Q code is capable of solving the Navier-Stokes equations
8. tighten analvsis convergence tolerance, g = _. with any of several turbulence models. The gradient
0<Z<I version of this code, which was used for aerodynamic
enddo sensitivity analysis, was generated I)3, an tmeonveTI-
tional application al of the autonmtic differentiation
This pseudocode is similar to that used in the code ADIFOR a-_,aa to produce a relatively efficient,
Biros and (;hattas 2_ SAND approach. Specifically, direct triode, gradient analysis code. CFLaD.ADII. a4
both approaches use a Sequential Quadratic Program- it should be l)ointed out that the ADIF()R process
ruing (SQP) method to solve the design equations produces a discretized derivative code consistent with
(st.ep 5) and an approximate factorization method to the discretized function analysis code. The addition of
solve the system equations (step 1). Step 3 above uses a stopping criterion based on the norm of the residual
an incremental iterative method with approximate of the field equations was the only moctification of the
factorization lo solve for derivatives in direct mode F L3I).A l)l Icode made to aceonmlodate 117¢'SASDO
rather that! as a solution of the adjoint equation of procedure.
(; ()F 11
AMERICAN INSTITUTE OF AEFIONAUTICS AN[) ASTRONAUTICS PAPE[¢ 21101 2527
FEM structural model were a subset, of the CFI) aero-
CFDmesh
C-Otopology dynamic surface mesh points (see Fig. 3) for the
73x25x25volume
present SASDO application. This lack of generality
49x25onthewing
allowed for simplifications in the data transfers and, al-
though an important issue, it was not deemed crucial
FEMmesh
for these initial SASDO demonstrations. Future ap-
3251elements:
2240CST plications to more complex configurations should allow
1011truss for transDr of conserved information between arbit rary
583points
meshes as required by the individual disciplines. A
recent review of such data transfer techniques and a
Fig. 3 Computational meshes for wing analysis. specific proposed one are given in Ref. 40.
Conventional (NAND) and SASDO (SAND) proce-
The surface geometry was geuerated based on tile dures were inlplemented using the SQP nlethod of the
paratneters described ill a previous section by a code DOT 41 optimization software. All computations were
utilizing the Hapid Aircraft. Parameterization Input executed on an S(;I Origin 2000 TM workstation with
Design (RAPID) technique developed t7)' Smith, et 250MHz R.10000 T'M processors. The CFD sensitiv-
al. 3s This code was preprocessed with ADIFOt/to gen- ity calculations were partitioned and run on several
erate a code capable of producing sensitivity deriva- processors to reduce required memory and elapsed op-
tives as well. timization time. This partitioning, however, resuhs in
Tile CFD volume mesh needed by the flow analysis additional accumulated colnputation time due to the
code was generated using a version of the CSCMDO a6 nature of ADIF'OR-generated .,sensitivity analysis code.
grid generation code. The associated grid sensitivity
Results
derivatives needed by the flow sensitivity analysis were
generated with all automatically differentiated ver- The ot)timization results shown in this work are
sion of CSCMI)O. ar Ill addition to the paranleterized for design problems involving only four or eight de-
surface mesh and accomt)anying gradients. CSCMDO sign variables out of tile 21 availal)le wing lTarameters.
requires a baseline volume inesh of similar shalle and The results shown by these authors in air earlier work _'
identical topology. The 45,000 grid point baseline vol- used design variables that directly affected either the
ume mesh of C-O topology used in the present, flexible aerodynamic analysis alone or both the aerodynamic
wing examples was obtained with the Gridgen TM code. analysis and the structural analysis. In this work. ad-
The wing surface portion of the mesh is shown in ditional design variables are chosen that directly affect
Fig. 3. This mesh is adnfittedly particularly coarse only the structural analysis. The flow conditions for
by current CFD analysis standards. the wing optimizatiolls were Mr, = 0.8 and a = 1°.
The structural analysis code as used to couipule
....... : i
the deflection of the elastic wing was a generic fi-
nite element code. The fexible structure for the wing
shown in Fig. 3 was discretized by 583 nodes; there
were 2,141 constant-strain triangle (CST) elements
and 1,I 10 truss elelnents. Zone boundaries for the de-
sign variables controlling element size are also shown in original
Fig. 3. Because the ela_st,ic deformation was assumed
to be small, linear elasticity was deemed to be ap-
propriate. The structural sensitivity equations were
derived based upon the direct differentiation method.
We note that the sensitivity of the aerodynamic forces
appears as a term on the right-hand side (I{ttS) of the Fig. 4 Comparison of planform shapes and sur-
deflection sensitivity equations. The derivative of the face pressure contours for 4-design-variable cases,
stiffness matrix in these sensitivity equations was also 3Ix = 0.8, _ -- 1°.
generated m' t73, using the ADIFOR a2'aa technique. We
note that the coefficient matrix of the structural sensi- Four-Design-Variable Problems
tivity equations was identical to that of the structural Table 1 and Fig. 4 show results of several optilniza-
equations. Consequently, these structural sensitivity tion problems revolving four design variables: the tip
equations were solved efficiently by backward substi- chord ct, the tip setback xt, and the structural element
tution with different RHSs for each sensitivity. size factor for the two most inboard zones, ['l and
At the wing surface, i.e., the interface where aero- 1"2. Two of the cases, designated (!onvc and SASDOc
dynamic load and structural deflection information is in Table 1, represent direct comparisons of SASDO
transferred, it was assulned that surface nodes of the and the conventional method for consistent accuracy
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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2001 2527
Table 1 Summary of Four-Design-Variable Cases.
baseline Optimizations
SASDOA ConvB SASDOc Convc
1.07778 1.07281 1.07827 1.07697
Ct 1.
1. 1.93259 1.91880 1,93052 1.92614
Xt
FI l. 1.06631 1.08008 1.09637 1.09982
1. 0.69510 0.69187 0.68664 0.68691
["2
F -7.149 -10.187 -10.181 -10.1_6 -10.183
gl -.0302 -.000760 -.00177 -.0000629 -.000915
g._, -.8882 -0.880 -0.876 -0.88(t -0.879
g3 -2647 -.000639 -.000159 -.000149 +.000108
c 9.6e-7 9.9e-5 8.9e-7 6.6e-6 4.0e-6
state 1 1.1 4.8 2.0 3.9
grad 27.1 63.1 26.9 40.4
of flmction and gradient analyses. The other two cases have been separated in Fig. 5. The total time for
show effects of changes in the accuracy of the function performing this optilnization t)roblem was reduced by
and gradient analyses. The resulting designs are es- 36 percent using the SASDO method. The analysis
sentially identical for all four cases. Figure 4 shows alone was reduced by 55 l)er(:ent, but the gradient eval-
a comparison of the wing planform and t,]le surface uation was the dominant cost,.
pressure coettlcient results for the baseline design and J
the design designated SASDOc m Table 1. The shock
wave has been weakened somewhat in the optilnized
cases from lhal on t,h(" original wing, as one would
eXl-,ecl. As one can see qualitatively in Fig. 4 and _<_DO
tmmerically from the values of the objective function
l:, lhe constraints 9i, and the final design variables
in Tab[e 1, the final designs are very similar for the
four problems. The relative computational costs of
5O
l 1
la
R d(FEM)
tOO d(CFD) Fig. 6 Comparison of planform shapes and sur-
face pressure contours for 8-design-variable cases,
® mesh
M.,_ =0.8, a = 1°.
-,4 _ dF(EmMesh)
la
III CFD
Eight-Design-Variable Problems
@
Table 2and Fig. 6show results of three optimization
problems involving eight design variables: the same set
0
used m the four-design-variable cases with the inclu-
conven- SASDO sion of the span b, the root section max camber zr,
tional
and the structural element size factor for two more
zones, F3 and F4- Two of the cases, designated ConvB
Fig. 5 Comparison of computation cost of four-
and SASDOB in Table 2, represent direct comparisons
design-variable optimization problem using the
conventional and SASDO methods. of SASDO and the conventional method for consis-
tent accuracy of the function and gradient analyses.
the optimizations are shown in Table 1 and Fig. 5. Figure 6 shows a comparison of wing planform and
The accunmlated function and gradient analysis times surface pressure coefficient results for the baseline de-
((tenoled as Zstate and £grad) are shown separately. sign and the design designated SASDOB in Table 2.
They have been normalized by the cost of the base- The relative computational costs of the optimizations
line COul)led function analysis. The components of the are shown in Table 2 and Fig. 7. The total time for
analyses were lumped together in Table 1 but they performing this optimization problem was reduced by
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AMEF_ICAN INSTITLITE OF' AERONAIr'I'I('S AND ASTRONAUTICS PAPEH 2001 2527