Unsteady Optimization Using a Discrete Adjoint Approach Applied to ...

clankflaxMechanics

Feb 22, 2014 (3 years and 5 months ago)

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Uns
teady
Opt
im
izat
io
n
Us
ing
a
D
i
scre
t
e
Adjo
in
t
Approa
c
h
Appl
ied
to
Aeroaco
ust
ic
Shap
e
D
esi
gn
Mark
us
P
.
R
umpf
k
eil
!
and
Da
vid
W.
Zing
g

Un
i
versi
t
y
of
T
o
r
ont
o
I
nstitute
for
A
er
osp
ac
e
Studies
4925
Du!erin
Str
e
et
,
T
or
onto,
O
N
,
M3H
5T6,
Canada
In
this
pa
p
er,
shap
e
op
ti
mizat
i
on
is
used
to
m
i
n
i
mize
aero
dyna
mi
c
n
o
ise
in
an
un
s
t
eady
tra
i
ling-edge

o
w.
F
irst,
a
generic
time-d
ep
end
en
t
opt
i
ma
l
d
es
ign
p
ro
b
lem
i
s
in
t
ro
d
uced
an
d
t
he
d
eriv
a
ti
on
of
t
he
d
is
crete
adjoi
n
t
equa
ti
ons
i
n
a
g
en
eral
ap
proac
h
is
outli
n
ed.
The
pr
e
sen
ted
framew
o
rk
i
s
then
a
pplied
to
a
time-
dep
en
den
t
lamina
r

o
w
pa
s
t
an
a
co
u
s
t
i
cally
compact
a
i
rfoi
l.
T
he
results
sho
w
a
si
gnifica
n
t
red
uction
of
u
p
to
94
p
ercen
t
in
th
e
total
rad
i
a
ted
a
c
ou
s
tic
p
o
w
er
wi
t
h
reasona
ble
co
mp
uta
ti
ona
l
c
ost
u
s
ing

f
teen
shap
e
d
esi
gn
v
ariables.
Nom
enclat
ure
a
!
F
re
e
s
tr
e
am
s
p
e
ed
of
soun
d
c
Chor
d
le
n
gth
¯
C
L
Mean
lif
t
c
o
e!c
ien
t
¯
C
D
Mean
dr
ag
co
e!
cien
t
I
n
Ob
j
e
ctiv
e
f
un
c
tion
at
ti
m
e
step
n
J
Ob
j
e
ctiv
e
f
un
c
tion
!
J
!
Y
Gr
adien
t
of
ob
jec
ti
v
e
fu
nction
M
!
F
re
e
s
tr
e
am
Mac
h
n
um
b
er
N
T
otal
n
u
m
b
e
r
of
time
s
teps
N
"
Num
b
e
r
of
coarse
time
ste
p
s
p
Pr
e
ss
u
re
Q
n
Flo
w
v
ar
iabl
e
s
at
ti
m
e
step
n
R
n
Unsteady
flo
w
res
id
ual
(
!
Q
n
R
n
)
T
T
ran
s
p
ose
of
th
e
un
s
teady

o
w
Jac
ob
ian
R
Flo
w
res
id
ual
R
e
Re
y
nold
s
n
u
m
b
e
r
S
Air
foil
surf
ac
e
t
Time
T
Fin
al
time
u
!
F
re
e
s
tr
e
am
v
e
lo
c
it
y
Y
Des
ign
v
ar
iabl
e
s
L
Lagran
gian
!
!
F
re
e
s
tr
e
am
densit
y
"
n
Adj
oin
t
v
ariab
les
at
time
s
tep
n
"
t
Time
di
s
cretiz
at
ion
s
t
e
p
"
t
"
Coarse
time
d
is
cretization
ste
p
I.
In
t
ro
ducti
on
and
Mo
ti
v
at
io
n
The
u
s
e
of
s
teady
-
state
aero
d
yn
am
ic
optimization
metho
d
s
in
th
e
com
p
utati
onal
flu
id
d
ynamics
(CFD)
com
m
un
it
y
is
fair
ly
w
e
ll
e
stabli
s
h
e
d
.
1–
4
In
p
articul
ar
the
use
of
adj
oin
t
metho
d
s
,
whic
h
h
as
b
e
en
p
ionee
r
e
d
b
y
Jame
son
5
for
ste
ad
y
aeron
auti
c
al
des
i
gn
op
timization,
has
p
ro
v
e
d
to
b
e
v
ery
b
eneficial
s
i
nce
its
c
ost
is
in
dep
end
e
n
t
of
t
he
n
u
m
b
e
r
of
des
ign
v
ar
iables.
The
app
lica
t
ion
of
n
ume
r
ic
al
op
timization
to
air
fr
am
e-
generated
n
ois
e,
h
o
w
e
v
er,
h
as
n
ot
rec
eiv
ed
as
m
u
c
h
atten
tion
,
b
ut
w
i
th
th
e
s
ign
ifi
c
an
t
quietin
g
of
m
o
dern
engi
nes
,
air
frame
noise
no
w
com
p
e
tes
with
engin
e
n
oise
.
6
Th
us
air
fr
am
e-
generated
noise
is
an
im
p
ortan
t
com
p
onen
t
of
th
e
total
n
oise
r
adiated
from
com
me
r
c
i
al
aircraf
t,
es
p
e
ciall
y
du
rin
g
aircraf
t
app
roac
h
and
lan
din
g,
when
engi
nes
op
e
r
ate
at
reduced
th
rust,
and
air
fr
am
e
com
p
onen
ts
(suc
h
as
h
igh
-
li
ft
device
s)
are
i
n
t
he
depl
o
y
ed
s
tate.
7
F
utu
re
F
ederal
Aviati
on
Admin
istration
n
ois
e
r
e
gu
lation
s
,
the
pro
jec
ted
gro
wth
in
air
tra
v
el
an
d
the
in
c
r
e
ase
in
p
opu
lation
densit
y
!
Ph.D
C
and
idate
,
Stu
den
t
Me
m
b
er
AI
AA,
m
arkus@
o
d
djob.ut
ias.u
toron
to
.ca

Professor,
T
ier
I
Ca
nad
a
Researc
h
Chai
r
in
C
omp
utat
iona
l
Aero
dynam
ics,
Asso
cia
te
F
ello
w
A
IAA,
h
ttp:
//go
ldfin
ger.u
tias.u
toro
n
t
o.ca/
"
dw
z/
1
o
f
9
Ame
rican
I
n
stitu
te
of
Aero
naut
ics
a
nd
Astr
onau
tics
46th AIAA Aerospace Sciences Meeting and Exhibit
7 - 10 January 2008, Reno, Nevada
AIAA 2008-18
Copyright © 2008 by David W. Zingg and Markus P. Rumpfkeil. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
near
air
p
or
ts
will
r
e
qu
ire
fu
ture
c
i
vil
air
c
raf
t
to
b
e
s
u
bstan
tiall
y
qui
e
ter
than
c
u
rren
t
ones
.
Con
s
equen
tl
y
,
th
e
attem
p
t
t
o
u
nd
e
rstand
and
redu
c
e
airf
rame
noi
s
e
h
as
b
e
com
e
an
imp
or
tan
t
res
earc
h
t
opic.
8
Op
timal
c
on
tr
ol
te
c
hn
iqu
e
s
f
or
un
s
teady

o
w
s
are
n
e
eded
in
o
r
der
to
b
e
able
to
re
d
uce
air
frame-
generated
noi
s
e.
Thi
s
pap
e
r
p
res
en
ts
a
ge
n
e
r
al
f
rame
w
ork
to
c
al
c
u
late
th
e
gradi
e
n
t
in
a
non
lin
e
ar
un
s
t
e
ad
y

o
w
en
vir
onme
n
t
via
t
he
discre
t
e
ad
join
t
me
th
o
d.
The
pr
e
se
n
te
d
fr
am
ew
ork
i
s
th
e
n
ap
plied
to
an
ae
r
o
dyn
amic
noi
s
e
redu
c
t
ion
prob
lem
in
v
olvi
ng
unsteady
laminar
trai
ling-edge
flo
w
similar
to
on
e
pr
e
se
n
t
e
d
b
y
Marsden
et
al.
9
I
I.
F
o
rm
ul
ati
on
o
f
t
he
D
iscre
te
T
im
e-dep
enden
t
O
pt
im
al
C
on
t
ro
l
Pro
blem
In
the
foll
o
win
g
w
e
ass
u
m
e
th
at
w
e
con
tr
ol
an
u
nste
ad
y
flo
w
i
n
the
ti
m
e
i
n
terv
al
[
0
,
T
]
with
an
in
itial
flo
w
soluti
on
Q
0
at
t
=
0.
In
this
se
ction
w
e
use
th
e
imp
licit
Eu
ler
time
-marc
h
ing
m
etho
d
to
d
is
cretize
th
e
go
v
e
r
nin
g
e
qu
ation
s
in
t
im
e.
Thi
s
is
not
a
res
tr
ic
t
ion,
s
in
c
e
it
is
strai
gh
tfor
w
ar
d
to
mo
dif
y
the
e
q
uation
s
to
use
an
y
other
ti
m
e-
marc
hin
g
m
eth
o
d
(e.g.
s
ee
th
e
Ap
p
end
ix
f
or
the
deriv
ati
on
with
t
he
sec
on
d-order
bac
kw
ards
di!e
r
e
n
c
e
(B
DF
2)
ti
m
e-m
ar
c
hi
ng
m
etho
d,
whic
h
is
u
s
ed
to
obtai
n
the
r
e
sult
s
).
W
e
in
tr
o
du
c
e
a
cos
t
f
unction
J
=
N
!
n
=1
I
n
(
Q
n
,
Y
)
,
(1)
where
the
fu
nction
I
n
=
I
n
(
Q
n
,
Y
)
d
e
p
e
n
ds
on
the
time
-dep
e
n
den
t
flo
w
soluti
on
Q
n
and
design
v
ari
ables
Y
.
N
can
b
e
calc
u
lated
fr
om
t
he
r
e
l
ation
T
=
N
"
t
,
w
h
e
r
e
"
t
is
the
c
hosen
time
d
isc
r
e
tization
s
tep.
W
e
then
ass
u
m
e
th
at
R
=
R
(
Q
n
,
Y
)
c
on
t
ains
t
he
spatial
ly
disc
r
e
ti
z
ed
c
on
v
e
ctiv
e
an
d
visc
ou
s
flu
xes
as
w
ell
as
th
e
b
oun
dary
cond
ition
s
and
th
at
R
n
(
Q
n
,
Q
n
!
1
,
Y
)
:
=
Q
n
!
Q
n
!
1
"
t
+
R
(
Q
n
,
Y
)
=
0
(2)
defin
e
s
im
p
licitly
th
e
time
-dep
e
n
den
t
flo
w
soluti
on
Q
n
for
n
=
1
,
.
.
.
,
N
.
It
do
e
s
n
ot
matter
ho
w
on
e
s
ol
v
e
s
equati
on
(2)
as
long
as
R
n
=
0
f
or
all
n
,
s
i
nce
thi
s
is
th
e
requ
irem
en
t
for
th
e
f
ollo
wing
deriv
ati
on.
The
task
of
min
im
i
z
in
g
th
e
c
ost
f
un
c
ti
on
J
sub
jec
t
to
R
n
=
0
for
all
n
can
n
o
w
b
e
wri
tte
n
as
an
un
c
on
s
t
rain
e
d
op
timiz
ati
on
pr
oblem
of
min
im
i
z
in
g
the
Lagr
angian
f
un
c
ti
on
L
=
N
!
n
=1
I
n
(
Q
n
,
Y
)
+
N
!
n
=1
(
!
n
)
T
R
n
(
Q
n
,
Q
n
!
1
,
Y
)
(3)
with
resp
ec
t
to
Q
1
,
.
.
.
,
Q
N
and
!
1
,
.
.
.
,
!
N
,
w
h
e
r
e
!
1
,
.
.
.
,
!
N
are
the
N
v
ec
tor
s
of
Lagran
ge
m
u
ltip
liers
.
A
nec
es
sary
c
on
di
tion
for
an
extremal
i
s
that
th
e
gradi
e
n
t
of
L
with
resp
ec
t
to
Q
1
,
.
.
.
,
Q
N
and
!
1
,
.
.
.
,
!
N
shoul
d
v
an
is
h
.
Sin
c
e
w
e
s
tar
t
with
Q
0
and
calculate
th
e
s
tates
Q
1
,
.
.
.
,
Q
N
usin
g
th
e
constrai
n
ts
giv
en
b
y
equati
on
(2),
w
e
e
n
s
u
re
that
"
!
n
L
=
0
for
n
=
1
,
.
.
.
,
N
aut
om
ati
c
all
y
.
The
Lagrange
m
u
ltip
liers
!
n
m
u
s
t
no
w
b
e
c
hose
n
s
u
c
h
th
at
"
Q
n
L
=
0
f
or
n
=
1
,
.
.
.
,
N
,
whic
h
leads
to
0
=
"
Q
n
I
n
+
(
!
n
)
T
"
Q
n
R
n
+
(
!
n
+1
)
T
"
Q
n
R
n
+1
for
n
=
1
,
.
.
.
,
N
!
1
(4)
0
=
"
Q
N
I
N
+
(
!
N
)
T
"
Q
N
R
N
.
(5)
This
c
an
b
e
wri
tte
n
equ
iv
ale
n
tly
as
!
N
=
!
"
(
"
Q
N
R
N
)
T
#
!
1
(
"
Q
N
I
N
)
T
(6)
!
n
=
!
"
(
"
Q
n
R
n
)
T
#
!
1
$
(
"
Q
n
I
n
)
T
+
(
"
Q
n
R
n
+1
)
T
!
n
+1
%
for
n
=
N
!
1
,
.
.
.
,
1
.
(7)
Si
nce
Q
1
,
.
.
.
,
Q
N
ha
v
e
b
ee
n
c
al
c
u
late
d
fr
om
th
e
cur
re
n
t
iterate
of
Y
,
th
e
Lagr
ange
m
ul
tipl
ie
r
s
!
n
can
b
e
c
alcul
ate
d
rec
u
rsiv
ely
b
ac
kw
ard
s
from
th
e
te
r
m
i
nal
b
ou
nd
ary
cond
ition
(6)
using
(7).
The
s
y
s
tem
of
equati
ons
(6)
and
(7)
is
kno
wn
as
the
syste
m
of
ad
join
t
equati
ons
for
th
e
m
o
del
(2),
or
as
the
ad
join
t
mo
del.
In
th
is
con
te
x
t,
the
Lagr
ange
m
ult
ipli
e
r
s
are
al
s
o
kno
wn
as
th
e
ad
join
t
v
ari
ables
.
Fin
ally
,
one
can
e
v
alu
ate
the
gr
adien
t
of
J
with
res
p
ec
t
to
the
d
e
sign
v
ariab
le
s
Y
,
w
h
ic
h
can
then
b
e
used
in
a
gradi
e
n
t
-
b
as
ed
opti
m
ization
al
gorith
m
suc
h
as
B
F
G
S
10
–1
3
to
fin
d
the
optim
um:
"
J
"
Y
=
"
L
"
Y
=
N
!
n
=1
"
Y
I
n
(
Q
n
,
Y
)
+
N
!
n
=1
(
!
n
)
T
"
Y
R
(
Q
n
,
Y
)
.
(8)
2
o
f
9
Ame
rican
I
n
stitu
te
of
Aero
naut
ics
a
nd
Astr
onau
tics
In
summ
ar
y
,
t
he
gr
adien
t
is
d
e
t
e
rmin
e
d
b
y
the
solu
tion
of
th
e
ad
join
t
equ
ations
in
rev
e
r
s
e
time
fr
om
th
e
terminal
b
ou
ndar
y
condi
tion
an
d
th
e
p
artial
d
e
r
iv
ativ
es
of
t
he
flo
w
r
e
sid
ual
an
d
ob
jec
t
iv
e
f
un
c
tion
with
res
p
ec
t
to
the
d
e
sign
v
ar
iabl
e
s
(whi
le
Q
n
is
held
c
on
s
t
an
t).
On
e
can
als
o
s
ee
t
hat
th
e
c
ompu
tation
al
cos
ts
of
un
s
t
e
ad
y
opt
im
i
z
ati
on
p
robl
e
ms
are
dir
e
ctly
prop
ort
ional
to
th
e
d
e
sired
n
um
b
er
of
ti
m
e
ste
p
s
an
d
(almos
t)
in
dep
end
e
n
t
of
t
he
n
u
m
b
e
r
of
des
ign
v
ar
iables.
I
I
I.
The
A
er
o
dy
nam
ic
Noi
se
Reduct
io
n
P
r
obl
em
W
e
no
w
pr
e
se
n
t
an
u
nste
ad
y
ae
r
o
dy
namic
n
oise
redu
c
t
ion
p
robl
e
m
whic
h
app
lies
th
e
ab
o
v
e
f
rame
w
ork
in
p
ractice
.
The
airf
oil
ge
ometry
,
whic
h
is
a
short
e
n
e
d
v
ersion
of
th
e
air
foil
u
s
ed
in
exp
e
r
ime
n
ts
b
y
Blak
e
,
14
is
s
h
o
wn
i
n
Figu
re
1.
T
h
is
geom
etry
i
s
v
ery
sim
i
lar
to
the
one
use
d
b
y
Marsden
et
al.
9
in
t
heir
noi
s
e
mini
m
ization
u
s
i
ng
a
surr
ogate
managem
en
t
fr
am
ew
or
k.
The
air
foil
c
hord
i
s
10
ti
m
es
its
th
ic
k
nes
s,
the
f
ree
stream
Mac
h
n
um
b
er
is
M
"
=
0
.
2
w
i
th
a
R
eyn
olds
n
um
b
er
of
R
e
=
10
,
000,
and
t
he
an
gle
of
attac
k
is
0
#
.
x
y
0
0
.
2
0
.
4
0
.
6
0
.
8
1
-
0
.
0
5
0
0
.
0
5
Figure
1
:
Bl
ak
e
airfoi
l
us
ed
i
n
unsteady
l
ami
nar
flo
w
p
r
o
bl
em
with
the
thic
knes
s
constrain
t
li
ne
(da
s
hed).
The
ri
gh
t
half
o
f
t
he
upp
er
surface
i
s
al
lo
w
ed
to
deform
and
the
fifteen
B-spl
ine
con
t
rol
p
oi
n
ts
w
hic
h
are
used
as
des
ign
v
a
ri
ables
are
sho
w
n
as
sq
uares.
F
or
un
s
t
e
ad
y
lamin
ar
flo
w
p
as
t
an
air
foil
at
lo
w
Mac
h
n
u
m
b
e
r
,
the
acoustic
w
a
v
ele
n
gth
asso
c
i
ate
d
with
th
e
v
or
te
x
s
h
e
d
din
g
is
t
ypi
c
all
y
lon
g
relativ
e
to
t
he
airf
oil
c
hord
.
9
The
noi
s
e
ge
n
e
r
ation
fr
om
suc
h
an
acoustic
al
ly
c
omp
ac
t
air
foil
can
b
e
exp
re
sse
d
u
s
i
ng
Curl
e
’s
e
x
te
n
s
i
on
to
the
Ligh
th
ill
th
e
or
y
15
and
a
c
ost
fu
nction
J
,
whic
h
is
p
rop
ortion
al
to
th
e
total
r
adiated
acoustic
p
o
w
er
c
an
b
e
d
e
r
iv
e
d
:
16
J
=
&
"
"
t
'
S
n
j
p
1
j
(
y
,
t
)
ds
(
2
+
&
"
"
t
'
S
n
j
p
2
j
(
y
,
t
)
ds
(
2
.
(9)
Here
,
p
ij
is
t
he
com
p
res
siv
e
stres
s
te
n
s
or
,
n
j
are
the
n
ormalize
d
comp
on
e
n
ts
of
th
e
out
w
ard
n
ormal
to
th
e
air
foil
s
u
rface
S
,
and
y
is
th
e
ai
rfoi
l
surf
ac
e
p
os
i
tion
v
e
ctor.
The
o
v
e
rb
ar
d
e
n
otes
t
im
e-a
v
eragin
g
o
v
er
th
e
c
h
os
en
time
in
terv
al,
an
d
rep
e
ated
in
dices
fol
lo
w
th
e
u
s
u
al
Ein
s
tein
summation
con
v
en
tion
.
The
radi
ation
in
thi
s
c
ase
i
s
of
d
ip
ole
t
y
p
e,
cause
d
b
y
the

uctuatin
g
li
ft
an
d
drag
force
s;
th
e
reader
is
r
e
f
e
rr
e
d
to
W
ang
et
al.
17
for
more
details
on
airfoi
l
s
elf-noi
s
e
d
ue
to
v
or
te
x
shedd
ing.
The
geom
etry
of
th
e
airf
oil
is
d
e
sc
r
ib
ed
w
i
th
cub
ic
B-
spli
ne
c
u
rv
es
,
4
whic
h
means
that
s
ome
of
th
e
y
-
co
ord
inates
of
th
e
B
-spli
ne
con
trol
p
oin
ts
in
the
upp
er
righ
t
h
alf
of
the
airf
oil
c
an
eas
i
ly
b
e
u
s
ed
as
shap
e
design
v
ariab
le
s
(s
ee
Fi
gure
1).
Si
nce
the
c
ost
of
our
ad
join
t
ap
proac
h
is
i
ndep
e
n
den
t
of
the
n
um
b
er
of
d
e
sign
v
ariab
les
,
w
e
d
e
cid
e
d
to
u
s
e
consid
e
rab
ly
m
or
e
s
h
ap
e
d
e
sign
v
ariab
le
s
t
han
the

v
e
that
Marsden
et
al.
9
could
a!ord
in
th
e
i
r
s
tu
dy
usin
g
a
s
u
rrogat
e
m
an
age
men
t
framew
ork
.
W
e
u
s
e

ftee
n
shap
e
d
e
sign
v
ariab
les
in
th
is
res
earc
h,
th
us
giv
ing
th
e
airf
oil
more
f
re
edom
in
th
e
design
s
p
ac
e
to
t
ak
e
the
m
ost
b
eneficial
shap
e
as
giv
en
b
y
th
e
BF
GS
opti
m
i
z
er.
18
,
1
9
Ho
w
ev
e
r
,
w
e
i
m
p
ose
thic
kness
c
on
s
tr
ain
ts
via
a
q
uadr
atic
p
e
n
alt
y
me
th
o
d
to
ensure
that
th
e
airf
oil
h
as
a
c
ertain
m
in
im
um
th
ic
k
nes
s.
W
e
ap
pl
y
the
same
minim
um
thi
c
kn
e
ss
as
imp
os
ed
b
y
Mar
s
d
e
n
et
al
.
,
w
h
ic
h
i
s
giv
e
n
b
y
a
straigh
t
li
ne
conn
e
cting
th
e
le
f
t
e
d
ge
of
the
d
e
f
ormation
r
e
gion
an
d
th
e
trail
ing
edge,
as
s
h
o
wn
in
Figur
e
1.
W
e
u
s
e
a
C-m
es
h
with
298
#
95
n
o
des
,
whic
h
is
a
go
o
d
compromise
b
et
w
ee
n
the
acc
u
racy
of
the
flo
w
soluti
on
an
d
the
com
p
utat
ional
e
!ort
requi
red.
In
ord
e
r
to
solv
e
th
e
und
e
r
lyin
g
t
w
o-
d
ime
n
s
i
onal
unsteady
com
p
res
sibl
e
thin
-la
y
e
r
Na
vier-Stok
es
equati
ons
in
n
on-di
m
ensional
f
orm
w
e
use
our

o
w
s
ol
v
e
r
P
R
OBE
20
with
th
e
s
ec
on
d-ord
e
r
acc
u
rate
BDF2
time
-marc
h
ing
me
th
o
d.
The
spati
al
d
isc
retization
of
th
e
s
t
e
ad
y

o
w
res
i
du
al
R
=
R
(
Q
n
,
Y
)
i
s
t
he
s
ame
as
that
use
d
in
AR
C
2D
.
21
It
consists
of
se
cond
-order
cen
te
r
e
d
-
d
i!e
r
e
n
c
e
op
e
r
ators
with
sec
on
d-
and
fou
rth
-
d
i!e
r
e
n
c
e
s
calar
artifi
c
i
al
di
s
sipat
ion.
W
e
use
an
in
e
x
ac
t
Ne
wton
s
tr
at-
egy
20
,
2
2
to
dr
iv
e
th
e
d
is
cretize
d
unsteady

o
w
res
i
dual
R
n
to
10
!
12
at
e
ac
h
time
step
n
.
The
m
ai
n
c
om-
3
o
f
9
Ame
rican
I
n
stitu
te
of
Aero
naut
ics
a
nd
Astr
onau
tics
p
onen
ts
of
th
is
strategy
in
c
l
ud
e
th
e
matri
x-
f
ree
generali
z
ed
min
im
u
m
r
e
sidu
al
(G
MRES
)
m
eth
o
d
23
and
an
in
c
ompl
e
te
lo
w
er-up
p
er
f
ac
tor
iz
at
ion
24
ILU(
k
)
ri
gh
t
pr
e
cond
ition
e
r
w
i
th
a

ll
lev
e
l
of
k
=
4
t
o
inexactly
solv
e
th
e
lin
e
ar
s
y
s
tem
,
w
h
ic
h
res
u
lts
from
app
lyin
g
Ne
wton
’s
metho
d
to
e
q
uation
(2).
Th
e
p
re
cond
ition
e
r
is
b
as
ed
on
a
first-order
app
ro
ximation
of
th
e
flo
w
Jacobi
an
m
atr
ix,
and
th
e
m
at
rix-v
ec
tor
p
ro
d
ucts
requi
re
d
at
eac
h
GM
R
E
S
iterati
on
are
f
orme
d
with
first-order
fin
ite
d
i!e
r
e
n
c
es
.
Th
e
non
-
d
ime
n
s
ion
alization
is
ac
com
p
lished
with
the
follo
win
g
s
calin
g
paramete
r
s
:
t
he
f
ree
stream
dens
i
t
y
#
"
,
the
airf
oil
c
hord
c
as
a
l
e
n
gth
sc
al
e
,
the
fr
e
e
stream
sp
ee
d
of
soun
d
a
"
as
a
v
e
lo
c
it
y
sc
ale,
an
d
c/a
"
as
a
time
s
cale.
Ou
r
Re
yn
olds
n
um
b
er
of
10
,
000
is
b
as
ed
on
t
he
fr
e
e
stream
v
e
l
o
c
i
t
y
u
"
and
th
e
c
h
ord
length
c
.
M
arsden
et
al.
used
a
v
e
r
y
similar
s
calin
g
to
pr
e
se
n
t
their
res
u
lts
,
althou
gh
they
used
u
"
as
the
v
e
l
o
c
i
t
y
s
cale.
In
order
to
c
on
v
ert
t
he
ob
jec
tiv
e
f
unction
v
alue
fr
om
our
s
calin
g
to
M
ars
d
e
n
’s
scaling
w
e
h
a
v
e
to
div
ide
it
b
y
(
M
"
)
6
,
an
d
w
e
ha
v
e
t
o
m
ult
iply
ou
r
non
-dime
n
s
i
onali
z
ed
ti
m
e
b
y
M
"
to
b
e
able
to
c
ompar
e
i
t
to
M
arsden’s
non
-dime
n
s
i
onali
z
ed
ti
m
e.
F
or
the
remaind
e
r
of
th
is
p
ap
er
w
e
w
i
ll
rep
or
t
al
l
our
res
u
lts
wit
h
Marsden’
s
s
calin
g
t
o
eas
e
compari
s
on
s
.
The
B
i
-
CGS
T
AB
al
gorith
m
25
is
use
d
to
s
olv
e
the
lin
e
ar
s
ystem
s
in
th
e
ad
join
t
e
qu
ation
s
with
an
absolu
te
con
v
e
r
ge
n
c
e
tolerance
of
10
!
6
and
righ
t
prec
on
di
tioni
ng
with
ILU(5)
is
app
lied
to
acc
elerate
con
v
ergence
.
W
e
fou
nd
Bi-CGST
AB
to
b
e
ab
out

ft
y
p
e
rcen
t
faste
r
i
n
s
ol
vin
g
th
e
u
nste
ad
y
adj
oin
t
equati
ons
th
an
GM
R
E
S,
whic
h
w
e
s
ti
ll
u
s
e
in
our
un
s
t
e
ad
y
flo
w
solv
es
as
me
n
tioned
ab
o
v
e
b
e
cause
there
are
no
s
ign
ifican
t
com
p
utati
onal
sa
v
ings
b
y
u
s
i
ng
Bi-C
G
ST
AB
for
the
few
l
inear
it
e
rat
ions
w
e
ha
v
e
to
use
p
e
r
non
linear
(ou
te
r
)
i
te
r
ation.
Ho
w
ev
er,
for
a
s
teady
-
state
ad
join
t
p
rob
le
m
B
i
-
CGS
T
AB
w
orks
not
nearly
as
w
ell
and
w
e
are
usin
g
GMRE
S
i
nste
ad
.
The
reas
on
for
th
is
i
s
most
lik
ely
ac
coun
ted
for
b
y
th
e
f
ac
t
that
(
"
Q
n
R
n
)
T
is
more
diagon
ally
d
om
i
nan
t
t
han
t
he
transp
os
e
of
th
e
steady

o
w
Jacobi
an
(
"
Q
R
)
T
du
e
t
o
the
e
xt
ra
terms
on
th
e
di
agonal,
whi
c
h
mak
e
s
th
is
matri
x
m
or
e
s
u
ited
for
th
e
use
of
Bi-CGST
AB.
W
e
also
foun
d
th
at
the
alge
b
raic
grid
m
o
v
em
en
t
algori
thm
use
d
b
y
Ne
mec
and
Zingg
26
is
not
capabl
e
of
dealin
g
with
th
e
o
cc
asional
fai
rly
large
shap
e
c
h
anges
.
Th
u
s
w
e
u
s
e
a
qu
as
i-lin
e
ar
elastic
i
t
y
me
sh
m
o
v
em
en
t
me
th
o
d
27
,
2
8
with
th
ree
in
c
reme
n
t
s
.
IV.
Resul
ts
The
lam
i
nar

o
w
arou
nd
the
or
igin
al
B
l
ak
e
airf
oil
exhi
bits
un
s
teady
v
or
te
x
shedd
ing,
wh
ic
h
l
e
ad
s
to
an
osc
il
latory
c
ost
fu
nction
as
s
h
o
w
n
i
n
Figur
e
2
u
s
i
ng
a
time
ste
p
size
of
"
t
=
0
.
005.
Th
e
agree
me
n
t
b
e
t
w
e
en
our
c
ost
f
unction
for
the
ori
ginal
Blak
e
air
foil
and
the
one
s
h
o
w
n
in
Marsden
et
al.
9
is
reas
on
ably
go
o
d
,
e
v
en
th
ough
ou
r
grid
is
ab
ou
t

v
e
times
c
oar
s
er.
3
4
5
6
7
8
9
10
0
0.1
0.2
0.3
0.4
0.5
0.6
t
J
Figure
2:
Instan
taneous
(t
hi
n
li
ne)
a
nd
ti
me-a
v
eraged
(t
hi
c
k
li
ne)
cost
function
for
the
original
Blak
e
airfoi
l
v
s.
t
i
me.
4
o
f
9
Ame
rican
I
n
stitu
te
of
Aero
naut
ics
a
nd
Astr
onau
tics
In
the
ac
tu
al
op
timiz
at
ion
r
un
s
w
e
use
th
e
disc
r
e
te
v
ersion
of
th
e
ti
m
e-a
v
eraged
cos
t
fu
nction
giv
e
n
b
y
equati
on
(9)
on
c
e
it
i
s
su#c
i
e
n
t
ly
con
v
e
r
ge
d
.
After
eac
h
s
h
ap
e
mo
difi
c
at
ion
th
e

o
w
solv
e
is
w
ar
m
started
fr
om
the
origi
nal
Blak
e
airf
oil
p
erio
d
ic
s
teady
s
tate
soluti
on
an
d
th
e

o
w
is
al
lo
w
ed
to
e
v
olv
e
for
s
ome
time
to
e
stabl
is
h
a
n
e
w
p
e
ri
o
dic
s
teady
state
b
efore
the
cost
f
un
c
ti
on
is
calculated
(com
p
are
wit
h
Figu
re
6).
W
e
“ju
mp”
o
v
e
r
th
is
unp
h
ysical
adj
usting
p
e
r
io
d
as
qu
ic
kl
y
as
p
os
sib
le
b
y
takin
g
a
b
igge
r
time
ste
p
"
t
$
=
0
.
01
for
the
fir
s
t
N
$
=
300
steps.
On
c
e
w
e
reac
h
our
d
e
sired
con
tr
ol
w
i
nd
o
w
[3
,
10]
(where
w
e
time
a
v
erage
the
ob
jec
t
iv
e
f
unction
),
w
e
u
s
e
a
smaller
time
ste
p
"
t
=
0
.
005
for
an
other
1400
s
teps,
f
or
a
total
of
N
=
1700
ste
p
s
c
o
v
e
r
ing
a
time
in
terv
al
of
[
0
,
10]
for
e
ac
h

o
w
s
olv
e.
T
h
e
c
or
res
p
ond
ing
adj
oin
t
equ
ations
r
e
sulti
ng
fr
om
a
v
ariab
le
time
ste
p
ar
e
giv
en
in
th
e
App
end
ix.
X
Y
0
.
4
0
.
5
0
.
6
0
.
7
0
.
8
0
.
9
1
-
0
.
0
5
0
0
.
0
5
J
=
1
.
3
3
E
-
5
J
=
1
.
0
7
E
-
5
J
=
3
.
0
0
E
-
5
J
=
8
.
5
2
E
-
6
Figure
3:
The
i
nitial
ai
rfoil
shap
es.
W
e
start
the
op
timization
pro
c
edu
re
fr
om
four
d
i!
eren
t
init
ial
shap
es
,
wh
ic
h
are
sho
wn
togeth
e
r
with
th
e
ir
ob
j
e
ctiv
e
f
un
c
ti
on
v
al
ues
(with
out
the
quad
ratic
p
enalt
y
f
or
thi
c
kn
e
ss
c
on
s
t
rain
t
vi
olation
)
i
n
Figu
re
3:
1.
The
origin
al
B
l
ak
e
airf
oil
(in
r
e
d
)
2.
The
airf
oil
defin
e
d
th
rou
gh
the
t
hic
knes
s
c
on
s
tr
ain
t
lin
e
(in
gr
e
en)
3.
The
airf
oil
th
at
res
u
lts
f
rom
s
ettin
g
all
fif
tee
n
d
e
sign
v
ar
iables
to
their
sp
ec
i
fied
up
p
er
b
oun
d
(in
blu
e
)
4.
The
air
foil
that
r
e
sults
fr
om
s
ettin
g
all
fif
tee
n
des
ign
v
ariab
le
s
to
th
e
i
r
s
p
ec
ifi
e
d
lo
w
er
b
ou
nd
(
in
b
lac
k)
The

rst
three
in
itial
shap
e
s
do
not
viol
ate
an
y
th
ic
kn
e
ss
constrain
ts
;
h
o
w
ev
e
r,
th
e
fou
rth
on
e
d
o
e
s.
0
2
4
6
8
10
12
14
16
18
20
22
0
0.2
0.4
0.6
0.8
1
Number of iterations
J


Original Blake
Thickness line
Upper bound
Lower bound
0
5
10
15
20
25
30
35
40
10
−1
10
0
10
1
10
2
10
3
Number of iterations
Gradient norm


Original Blake
Thickness line
Upper bound
Lower bound
Figure
4:
Con
v
ergence
hi
stories
of
the
aeroacoustic
shap
e
de
sign
problem
s
usi
ng
fifteen
desi
gn
v
ariable
s.
5
o
f
9
Ame
rican
I
n
stitu
te
of
Aero
naut
ics
a
nd
Astr
onau
tics
The
c
on
v
ergence
h
istories
of
thes
e
ae
r
oac
ou
s
tic
s
h
ap
e
design
p
robl
e
ms
ar
e
sho
wn
in
F
igur
e
4.
T
h
e
ob
jec
t
iv
e
fu
nction
s
are
alw
a
ys
sc
aled
with
th
e
ini
tial
ob
jec
tiv
e
fun
c
ti
on
v
al
ue
of
th
e
origi
nal
Blak
e
airf
oil
J
0
=
1
.
33

10
!
5
to
mak
e
compari
s
on
s
eas
ier.
On
e
can
se
e
th
at
all
ob
j
e
ctiv
e
fun
c
t
ions
are
dri
v
e
n
to
m
u
c
h
sm
all
e
r
v
al
ues
in
ab
ou
t
t
w
o
t
o
e
igh
t
des
ign
iteration
s
an
d
th
at
the
impr
o
v
eme
n
t
after
that
is
on
ly
margin
al.
St
artin
g
from
th
e
or
iginal
B
l
ak
e
air
foil
le
ad
s
to
t
he
b
es
t
ai
rfoi
l
in
te
r
m
s
of
total
radi
ate
d
ac
ou
s
ti
c
p
o
w
er.
The
redu
c
ti
on
is
ab
out
94
p
e
rcen
t
fr
om
the
in
itial
v
al
ue
and
th
u
s
m
uc
h
l
arge
r
than
th
e
80
p
erce
n
t
ac
h
ie
v
ed
b
y
M
arsden
et
al
.
9
usin
g
fiv
e
d
e
sign
v
ar
iabl
e
s.
The
gr
adien
t
n
orms
are
onl
y
reduced
b
y
on
e
to
t
w
o
or
ders
of
magnitu
de,
imply
ing
that
th
e
opti
m
i
z
er
did
n
ot
fu
lly
c
on
v
e
r
ge
du
e
to
s
tal
ls
in
th
e
lin
e
se
ar
c
h
algor
ithm.
X
Y
0
.
4
0
.
5
0
.
6
0
.
7
0
.
8
0
.
9
1
-
0
.
0
5
0
0
.
0
5
J
=
7
.
6
4
E
-
7
J
=
8
.
6
9
E
-
7
J
=
9
.
0
6
E
-
7
J
=
9
.
3
7
E
-
7
Figure
5:
Final
i
mpro
v
ed
ai
rfoil
s
ha
p
es
(sol
id)
and
ini
t
i
al
airfoi
l
shap
es
(
das
hed)
.
Figu
re
5
sho
ws
t
he

nal
i
m
p
ro
v
e
d
airf
oil
s
h
ap
e
s
toge
th
e
r
w
i
th
their
ob
j
e
ctiv
e
fun
c
t
ion
v
alu
e
s
(this
time
with
the
qu
adr
atic
p
enal
t
y
f
or
th
ic
kn
e
ss
c
on
s
t
rain
t
violati
on
i
nclud
e
d
),
w
h
ic
h
are
v
ery
in
teres
ti
ng
an
d
com-
pl
e
tely
u
nexp
ecte
d
.
Th
e
in
c
r
e
ase
in
t
he
tr
ailin
g-e
d
ge
angl
e
to
dec
r
e
ase
th
e
trai
lin
g-
edge
noise
w
as
also
foun
d
b
y
Mar
s
d
e
n
et
al.
and
w
as
th
e
or
e
ti
c
all
y
p
redicted
b
y
Ho
w
e
29
for
t
urb
ulen
t
flo
w.
Ho
w
e
v
er,
the
“w
a
vy”
par
t
of
the
airf
oil
i
s
a
no
v
e
l
res
u
lt
an
d
to
t
he
b
est
of
t
he
auth
ors’
kn
o
wledge
h
as
on
ly
b
ee
n
rep
or
te
d
b
y
Ru
m
p
fk
eil
and
Zingg
30
,
3
1
in
a
pr
e
vi
ous
s
tu
dy
.
Pres
u
mably
M
arsden
et
al
.
di
d
n
ot
obtai
n
sim
i
lar
“w
a
vy”
shap
e
s
du
e
to
the
fact
that
th
e
y
u
s
ed
onl
y

v
e
d
e
sign
v
ari
ables
and
th
us
did
not
giv
e
t
heir
opti
m
i
z
er
e
n
ough
f
ree
d
om
to
c
ome
u
p
with
th
e
se
no
v
el
s
hap
es
.
Initi
al
Impro
v
ed
¯
C
L
¯
C
D
¯
C
L
/
¯
C
D
¯
C
L
¯
C
D
¯
C
L
/
¯
C
D
Ori
ginal
Bl
a
k
e
0
.
284
0
.
076
3
.
75
0
.
279
0
.
049
5
.
72
Thi
c
kne
ss
li
ne
0
.
265
0
.
054
4
.
95
0
.
279
0
.
049
5
.
67
Upp
er
b
ound
0
.
134
0
.
119
1
.
12
0
.
276
0
.
049
5
.
66
Lo
w
e
r
b
ou
nd
0
.
305
0
.
055
5
.
57
0
.
279
0
.
049
5
.
66
T
ab
l
e
1:
A
compari
son
o
f
the
m
ean
li
ft
and
drag
c
o
e!
c
i
en
ts
for
t
he
i
n
i
tial
and
im
pro
v
ed
ai
rfoil
s.
A
com
p
arison
of
t
he
mean
lif
t
and
drag
co
e#
cien
ts
f
or
th
e
in
itial
an
d
imp
ro
v
e
d
airf
oils
i
s
d
is
p
la
y
e
d
in
T
ab
le
1.
W
e
do
not
ha
v
e
to
ad
d
a
lif
t
constrain
t
or
a
p
e
n
alt
y
f
or
d
e
creas
ed
lif
t
to
the
ob
jec
t
iv
e
fu
nction
since
th
e
me
an
li
ft
c
o
e
#c
i
e
n
t
s
for
al
l
i
m
p
ro
v
e
d
airf
oils
eith
e
r
sta
y
ab
ou
t
the
same
or
in
c
rease
in
com
p
arison
to
their
ini
tial
v
al
ues
.
Th
e
m
ean
dr
ag
co
e
#cie
n
ts
are
d
e
creas
ed
in
all
c
ase
s.
This
m
eans
th
e
opti
m
izer
h
as
not
onl
y
pr
o
du
c
ed
ae
r
oac
ou
s
tically
impr
o
v
ed
airf
oils,
bu
t
also
as
a
b
ypr
o
du
c
t
the
in
itial
airf
oils
h
a
v
e
b
e
en
aero
d
ynamically
enhan
c
ed.
The
time
h
istories
of
C
L
and
C
D
for
the
origi
nal
Blak
e
airf
oil
b
e
f
ore
and
after
th
e
op
timiz
at
ion
ar
e
sho
wn
in
Figu
re
6.
O
ne
c
an
cle
ar
ly
se
e
th
e
unp
h
ysical
adj
ustin
g
p
e
ri
o
d
f
or
t
he
impro
v
ed
ai
rfoi
l
in
the
time
in
terv
al
[0
,
3]
b
efore
it
re
ac
hes
i
ts
new
some
what
p
erio
dic
ste
ad
y
state
.
A
redu
c
ed
mean
d
rag
as
w
ell
as
r
e
d
uce
d
osc
il
lation
ampl
itud
e
s
for
the
impro
v
ed
airf
oil
are
al
s
o
visib
le
.
W
e
also
t
ried
to
s
a
v
e
c
ompu
tation
al
time
and
s
tor
age
b
y
sa
ving
th
e

o
wfi
e
ld
in
th
e
con
trol
win
do
w
on
ly
ev
e
ry
s
econd
time
step.
W
e
ha
v
e
use
d
th
is
appr
oac
h
v
e
r
y
succ
es
sfu
lly
in
previou
s
s
tu
dies
.
30
,
3
2
Ho
w
ev
e
r
,
in
th
is
cas
e
th
is
ap
proac
h
do
es
n
ot
w
ork
v
e
r
y
w
ell,
sin
c
e
th
e
opti
m
i
z
er
is
bar
e
ly
ab
le
to
impro
v
e
th
e
in
itial
air
foils
e
v
en
s
l
igh
tly
with
t
his
i
nexac
t
gr
adien
t
inf
ormation
.
6
o
f
9
Ame
rican
I
n
stitu
te
of
Aero
naut
ics
a
nd
Astr
onau
tics
0
1
2
3
4
5
6
7
8
9
10
0.15
0.2
0.25
0.3
0.35
0.4
t
C
L
0
1
2
3
4
5
6
7
8
9
10
0.04
0.05
0.06
0.07
0.08
t
C
D
Figure
6:
T
im
e
hi
stories
of
C
L
and
C
D
for
the
o
ri
ginal
Blak
e
airfoil
b
efore
a
nd
aft
e
r
opt
i
mi
zat
i
on.
T
he
hi
sto
ri
es
of
the
i
niti
a
l
(
dashe
d)
and
i
mpro
v
ed
(sol
id)
ai
rfoil
s
vs
.
time
("
t
=
0
.
005)
are
sho
w
n.
V.
C
oncl
us
i
on
The
di
s
crete
adjoi
n
t
me
t
ho
d
w
as
succ
es
sfu
lly
app
lied
to
un
s
teady
laminar
trail
ing
e
d
ge
op
timization
res
u
ltin
g
in
a
s
i
gnifi
c
an
t
reduction
in
the
total
r
adiated
ac
ou
s
tic
p
o
w
er.
The
res
u
ltin
g
impr
o
v
ed
ai
rfoi
ls
sho
w
cas
e
v
e
ry
in
te
r
e
stin
g
an
d
c
ompletely
un
e
xp
ec
ted
s
h
ap
es,
t
here
b
y
sho
win
g
th
e
p
o
w
e
r
of
n
ume
r
ic
al
shap
e
opt
im
i
z
ati
on,
whic
h
c
an
lead
to
c
ou
n
terin
tu
itiv
e
re
sul
ts
.
It
w
ou
ld
b
e
v
e
r
y
i
n
te
r
e
stin
g
to
se
e
th
e
impr
o
v
ed
shap
e
s
teste
d
in
a
wind
tu
nn
e
l
to
c
on
firm
that
the
predi
c
ted
redu
c
tion
i
n
total
radi
ate
d
acoustic
p
o
w
er
is
ac
h
iev
e
d
in
r
e
ali
t
y
.
The
ge
n
e
r
al
fr
am
ew
or
k
p
re
sen
te
d
to
deri
v
e
th
e
u
nste
ad
y
d
is
crete
adjoi
n
t
equati
ons
for
opt
im
al
con
tr
ol
c
an
al
s
o
b
e
used
for
man
y
oth
e
r
i
nh
e
ren
tl
y
unsteady
op
timiz
ati
on
pr
oblems
.
Ou
r
fu
tur
e
w
ork
will
fo
c
u
s
on
the
abili
t
y
t
o
mo
d
ify
a
high
-lift
air
foil
confi
gurat
ion
to
reduce
th
e
rad
iated
noise
whi
le
main
tain
in
g
go
o
d
ae
r
o
dyn
amic
p
e
r
formance.
App
endi
x
In
t
his
app
endi
x,
w
e
deriv
e
the
d
is
crete
adjoi
n
t
equati
ons
in
the
for
m
i
n
whi
c
h
w
e
use
t
hem
to
pres
en
t
our
re
sul
ts
.
W
e
w
armstart
ou
r
flo
w
s
olv
e
at
t
=
0
whic
h
impli
e
s
that
w
e
kn
o
w
Q
0
and
Q
!
1
.
W
e
also
w
an
t
to
“j
ump”
o
v
er
the
adj
ustin
g
p
e
ri
o
d
as
q
uic
kly
as
p
oss
ib
le
th
u
s
t
aking
a
bi
gge
r
time
s
tep
"
t
$
for
N
$
time
ste
p
s
.
On
c
e
w
e
reac
h
the
d
om
ai
n
wh
e
r
e
w
e
actuall
y
w
an
t
to
c
on
trol
the
p
robl
e
m
w
e
use
a
s
mall
e
r
time
s
tep
"
t
for
N
!
N
$
time
s
t
e
p
s
.
Th
u
s
w
e
h
a
v
e
a
total
of
N
time
s
t
e
p
s
an
d
to
k
ee
p
th
e
s
ec
on
d-ord
e
r
time
ac
cur
ac
y
,
th
e
time
-dep
e
n
den
t
flo
w
solu
tion
Q
n
is
i
m
p
licitly
d
e

ned
thr
ough
th
e
fol
lo
w
i
ng
un
s
teady
residu
als
:
R
n
(
Q
n
,
Q
n
!
1
,
Q
n
!
2
,
Y
)
:=
3
Q
n
!
4
Q
n
!
1
+
Q
n
!
2
2"
t
$
+
R
(
Q
n
,
Y
)
=
0
for
n
=
1
,
.
.
.
,
N
$
R
N
!
+1
(
Q
N
!
+1
,
Q
N
!
,
Q
N
!
!
1
,
Y
)
:=
(2"
t
"
t
$
+
"
t
$
2
)
Q
N
!
+1
!
("
t
+
"
t
$
)
2
Q
N
!
+
"
t
2
Q
N
!
!
1
"
t
"
t
$
("
t
+
"
t
$
)
+
R
(
Q
N
!
+1
,
Y
)
=
0
7
o
f
9
Ame
rican
I
n
stitu
te
of
Aero
naut
ics
a
nd
Astr
onau
tics
R
n
(
Q
n
,
Q
n
!
1
,
Q
n
!
2
,
Y
)
:=
3
Q
n
!
4
Q
n
!
1
+
Q
n
!
2
2"
t
+
R
(
Q
n
,
Y
)
=
0
for
n
=
N
$
+
2
,
.
.
.
,
N
.
The
pr
oblem
of
m
in
imiz
i
ng
th
e
disc
r
e
te
ob
j
e
ctiv
e
fu
nction
giv
e
n
b
y
J
=
)
N
n
=
N
!
+1
I
n
(
Q
n
,
Y
)
i
s
th
e
n
equiv
alen
t
to
the
unconstrai
ned
optimization
p
rob
le
m
of
m
i
nimizing
th
e
Lagran
gian
fu
nction
L
=
N
!
n
=
N
!
+1
I
n
(
Q
n
,
Y
)
+
N
!
n
=1
(
!
n
)
T
R
n
(
Q
n
,
Q
n
!
1
,
Q
n
!
2
,
Y
)
with
r
e
sp
ec
t
to
Q
1
,
.
.
.
,
Q
N
and
!
1
,
.
.
.
,
!
N
.
This
l
e
ad
s
to
the
f
ollo
wing
equati
ons
f
or
!
n
:
0
=
(
!
n
)
T
"
Q
n
R
n
+
(
!
n
+1
)
T
"
Q
n
R
n
+1
+
(
!
n
+2
)
T
"
Q
n
R
n
+2
for
n
=
1
,
.
.
.
,
N
$
0
=
"
Q
n
I
n
+
(
!
n
)
T
"
Q
n
R
n
+
(
!
n
+1
)
T
"
Q
n
R
n
+1
+
(
!
n
+2
)
T
"
Q
n
R
n
+2
for
n
=
N
$
+
1
,
.
.
.
,
N
!
2
0
=
"
Q
N
"
1
I
N
!
1
+
(
!
N
)
T
"
Q
N
"
1
R
N
+
(
!
N
!
1
)
T
"
Q
N
"
1
R
N
!
1
0
=
"
Q
N
I
N
+
(
!
N
)
T
"
Q
N
R
N
,
whic
h
c
an
b
e
wr
itten
e
qu
iv
alen
tly
as
!
n
=
*
+
+
+
,
+
+
+
-
!
"
(
"
Q
n
R
n
)
T
#
!
1
$
(
"
Q
n
I
n
)
T
%
for
n
=
N
!
"
(
"
Q
n
R
n
)
T
#
!
1
$
(
"
Q
n
I
n
)
T
+
(
"
Q
n
R
n
+1
)
T
!
n
+1
%
for
n
=
N
!
1
!
"
(
"
Q
n
R
n
)
T
#
!
1
$
(
"
Q
n
I
n
)
T
+
(
"
Q
n
R
n
+1
)
T
!
n
+1
+
(
"
Q
n
R
n
+2
)
T
!
n
+2
%
for
n
=
N
!
2
,
.
.
.
,
N
$
+
1
!
"
(
"
Q
n
R
n
)
T
#
!
1
$
(
"
Q
n
R
n
+1
)
T
!
n
+1
+
(
"
Q
n
R
n
+2
)
T
!
n
+2
%
for
n
=
N
$
,
.
.
.
,
1
A
littl
e
c
ar
e
m
u
s
t
b
e
tak
e
n
i
n
c
al
c
u
latin
g
d
e
ri
v
at
iv
e
s
of
R
N
!
+1
with
resp
ec
t
to
Q
n
since
th
e
factors
i
n
fron
t
of
Q
N
!
+1
,
Q
N
!
and
Q
N
!
!
1
di
!
er
sligh
tl
y
from
th
e
usual
sc
h
e
me.
The
gr
adien
t
of
J
with
r
e
sp
e
ct
to
the
d
e
sign
v
ariab
les
Y
is
t
hen
giv
e
n
b
y
"
J
"
Y
=
"
L
"
Y
=
N
!
n
=
N
!
+1
"
Y
I
n
(
Q
n
,
Y
)
+
N
!
n
=1
(
!
n
)
T
"
Y
R
(
Q
n
,
Y
)
.
Ac
kno
w
le
dgm
en
ts
The
fu
nd
ing
of
th
e
se
cond
aut
hor
b
y
th
e
Natu
ral
Scie
n
c
es
and
En
ginee
r
ing
Res
earc
h
Cou
ncil
of
C
an
ada
and
the
Can
ada
Re
se
ar
c
h
Chair
s
pr
ogram
i
s
gratefu
lly
ac
k
no
wle
d
ge
d
.
Refer
ence
s
1
Ob
a
y
ashi,
S
.,
A
e
r
o
dynamic
Op
timizat
ion
with
Evo
l
ut
i
o
n
a
ry
A
l
go
rith
ms
,
In
v
erse
Design
an
d
O
ptim
izat
ion
Metho
ds
,
Lectur
e
Serie
s
1
997-
05,
e
dited
b
y
R
.
A.
V.
den
Br
aem
bus
sc
h
e
and
M.
Man
na,
V
on
Ka
rman
Inst
itute
for
Flu
id
Dynam
ics,
Brus
sels,
1
997.
2
Ja
mes
on,
A.,
P
ie
rce,
N.
A.,
and
Ma
rtine
lli,
L.,

Opti
m
u
m
Aero
dyna
mic
D
esign
U
si
ng
th
e
Na
vie
r-Sto
k
es
Eq
u
atio
ns,”
The
o
r
et
ic
al
and
Com
puta
tional
Flui
d
Dyn
a
mics
,
V
ol.
1
0,
No.
1,
19
98,
p
p.
2
13–2
37.
3
Ande
rson,
W.
K
.
and
Bon
haus
,
D.
L
.,
“Airfoil
Desig
n
on
Unstru
cture
d
G
rid
s
for
T
urbu
len
t
Flo
ws,”
AI
AA
J
ournal
,
V
o
l.
37
,
No.
2
,
199
9,
pp
.
18
5–19
1.
4
Neme
c,
M.
and
Z
ingg,
D.
W.
,
“Newt
on-K
rylo
v
Algori
thm
for
Aero
dynam
ic
Desig
n
Usi
ng
the
Na
v
i
er-St
ok
es
Eq
u
atio
ns,”
AI
AA
Jour
n
a
l
,
V
ol.
4
0,
No.
6,
20
02,
pp.
1
146–
1154
.
5
Ja
mes
on,
A.,

Opt
im
um
A
e
ro
d
ynami
c
Des
ign
Usi
ng
C
on
trol
The
ory,”
Com
puta
tional
Flu
id
Dyn
a
mics
R
eview,
H
afez,
M.,
Oshima
,
K.
(e
d
s)
,
Wil
ey:
New
Y
ork,
4
95-5
28,
1
995.
6
Sin
ger,
B.
A.,
Bren
tne
r,
K
.
S.,
and
Lo
c
k
ard
,
D
.
P
.,

Com
puta
tion
al
A
e
roaco
ustic
Ana
ly
s
is
o
f
Sla
t
T
ra
iling
-E
d
ge
Flo
w,”
AI
AA
Jour
n
a
l
,
V
ol.
3
8,
No.
9,
20
00,
p
p.
1
558–
1564
.
7
Kh
orram
i,
M.
R.,
Berkma
n,
M.
E
.,
and
Cho
udha
ri,
M.,
“Uns
tead
y
Flo
w
Com
puta
tion
s
of
a
Slat
with
a
Blun
t
T
rai
ling
Edge,”
AI
AA
Jour
n
a
l
,
V
ol.
3
8,
No.
11,
2
000,
pp.
2050
–205
8.
8
Sin
ger,
B.
A.
a
nd
Guo
,
Y.,
“De
v
elo
pmen
t
of
Com
puta
tion
al
Aeroa
cou
stics
T
o
ols
f
o
r
Airf
r
ame
N
o
ise
C
alcu
latio
ns,”
Inte
rnatio
n
a
l
Jo
urnal
of
Comp
uta
tional
Fl
uid
Dyn
a
mics
,
V
ol.
1
8(6)
,
200
4,
pp
.
45
5–46
9.
9
Marsd
en,
A.
L.,
W
ang,
M.,
D
enni
s
Jr.
,
J.
E.,
an
d
Moin
,
P
.,
“O
ptim
al
Aero
acou
stic
S
hap
e
D
esign
Using
the
S
urrog
ate
Mana
gem
en
t
F
rame
w
o
rk
,

Opt
imizat
i
o
n
a
n
d
E
ngin
e
e
rin
g
,
V
ol.
5
(2),
2004
,
pp.
235
–262
.
10
Bro
yden,
C
.
G.,
“Th
e
Co
n
v
erge
nce
o
f
a
C
lass
o
f
Doubl
e-Rank
Min
imiza
tion
Algori
thms
,”
Jo
u
r
n
a
l
Inst.
Ma
th
.
Ap
pli
c
.
,
V
o
l.
6,
1970
,
pp
.
76–
90.
8
o
f
9
Ame
rican
I
n
stitu
te
of
Aero
naut
ics
a
nd
Astr
onau
tics
11
Flet
c
h
er,
R.,

A
N
e
w
App
roac
h
t
o
Varia
ble
Me
tric
Alg
orith
ms,”
Com
pute
r
Jour
n
a
l
,
V
ol.
1
3,
19
70,
pp.
3
17–3
22.
12
Goldfarb,
D.,
“A
Famil
y
of
Vari
able
Metri
c
Up
dat
es
D
eriv
e
d
b
y
Var
iatio
nal
Means
,”
Mat
hem
atics
o
f
Co
mput
i
ng
,
V
o
l.
24,
19
70,
p
p.
23
–26
.
13
Sh
anno
,
D.
F.,
“Co
ndit
ionin
g
of
Qua
si-Newt
on
Meth
o
d
s
f
o
r
Funct
ion
Min
imiza
tion
,”
Mat
hem
atics
of
Co
mput
i
ng
,
V
ol
.
24,
19
70,
p
p.
64
7–6
56.
14
Blak
e,
W.
K.,
“A
Stati
stica
l
De
scrip
tion
of
P
re
ssure
an
d
V
elo
cit
y
Fields
at
t
he
T
railin
g
E
dg
e
of
a
Fla
t
Strut
,”
DTNSRDC
Rep
ort
4
241,
Da
v
id
T
a
ylo
r
Na
v
al
Shi
p
R
&
D
Cen
ter
,
Beth
esda
,
Marylan
d,
19
75.
15
Cu
rle,
N.,
“Th
e
Influ
ence
of
Soli
d
Bou
nda
ry
up
on
Aero
d
y
n
maic
Sou
nd,”
P
ro
c.
R
o
y
a
l
So
c.
L
on
d.
A,
2
31:5
05-5
14,
1
955.
16
Marsd
en,
A.
L.,
W
an
g,
M.,
a
nd
Kou
mou
tsak
os,
P
.,

Opti
mal
Aero
acou
stic
Sh
ap
e
Desig
n
using
Ap
pro
x
i
mati
on
Mo
delin
g,”
Ann
ual
R
e
searc
h
B
riefs,
Ce
n
t
er
for
T
urb
ulen
ce
Resea
rc
h
,
St
anford
U
n
iv
ersi
t
y
,
200
2.
17
W
a
ng,
M.,
Lele
,
S.
K.,
an
d
Mo
in,
P
.,

Com
puta
tion
of
qua
drup
ole
no
ise
u
sing
aco
ustic
an
alogy,”
AI
AA
Jo
urnal
,
V
o
l.
3
4,
No.
1
1,
1
996,
pp.
2
247
–225
4.
18
Byrd,
R.
H.,
L
u
,
P
.,
No
ced
al,
J
.,
and
Z
h
u
,
C.,

A
Limit
ed
Memo
ry
Algo
rithm
for
Bou
nd
Con
strai
ned
Op
timi
zatio
n,”
SIAM
J.
Scientifi
c
Comp
utin
g
16
,
V
ol.
5
,
199
5,
p
p.
11
90–1
208.
19
Zh
u,
C.,
Byrd,
R.
H.,
L
u
,
P
.,
and
N
o
ce
dal,
J.,
“L-BF
GS-B
:
A
L
im
ited
Memo
ry
F
OR
T
R
AN
Co
d
e
for
Sol
v
i
ng
Bou
nd
Co
nstra
ined
Opt
imiza
tion
P
ro
blem
s,”
T
e
c
h
.
R
e
p.
NAM-11
,
E
ECS
Dep
artme
n
t
,
North
w
est
ern
Uni
v
ersit
y
,
1994
.
20
Puey
o
,
A.
an
d
Zi
ngg,
D.
W.,
“E!ci
en
t
Newto
n-K
ry
l
o
v
So
lv
er
f
or
Aero
dynam
ic
Co
mpu
tati
ons,”
AI
AA
Jour
n
a
l
,
V
ol.
3
6,
No.
1
1,
1
998,
pp.
1
991
–199
7.
21
Pullia
m,
T.
H.,
E!
cie
n
t
S
o
l
ut
ion
Meth
o
ds
for
th
e
N
avie
r-S
t
oke
s
Equat
ion
s
,
L
e
cture
Note
s
for
the
V
o
n
Karm
an
I
n
stitu
te
F
or
Flu
id
Dyna
mics
L
ec
ture
Serie
s,
198
6.
22
Isono
,
S.
and
Zing
g,
D.
W.,
“A
Run
ge-K
utta-
N
e
wto
n-Krylo
v
Algo
rithm
for
F
ourth
-Ord
er
Imp
licit
Tim
e
Marc
hin
g
App
lied
to
U
n
stea
dy
Flo
ws,”
AI
AA,
2
004-0
433
,
200
4.
23
Sa
ad,
Y.
and
Sc
h
ultz
,
M.
H.,
“GMR
ES:
A
G
en
eraliz
ed
Min
ima
l
Residu
al
Algo
rithm
for
S
olving
N
o
nsymm
etric
L
i
near
Syste
ms,”
SIAM
Jo
urnal
on
S
cie
n
tific
and
Sta
tistic
a
l
Co
mput
i
ng
,
V
ol.
7
No.
3,
198
6,
pp
.
856
–86
9.
24
Meijerin
k
,
J.
A.
a
nd
v
a
n
d
er
V
orst,
H
.
A.,

An
Itera
tiv
e
Sol
ution
Met
ho
d
for
L
in
ear
System
s
o
f
whic
h
the
Co
e!
cien
t
Matri
x
is
a
S
y
m
me
tric
M-Ma
trix,”
Mat
hem
atics
o
f
Comp
utat
ion
,
V
ol.
V
ol.
3
1,
No.
137,
1977
,
pp
.
148
–162
.
25
v
an
de
r
V
orst
,
H.,
“Bi-C
G
ST
AB:
A
F
a
st
and
Smo
ot
hly
Con
v
e
rging
V
arian
t
o
f
Bi-
CG
for
th
e
Solu
tion
of
Non
symme
tric
Linear
System
s,”
SIAM
Jo
urnal
on
S
cie
n
tific
and
Sta
tistic
a
l
Co
mput
i
ng
,
V
ol.
1
3,
19
92,
pp.
6
31–6
44.
26
Nem
ec,
M.
and
Z
ingg
,
D.
W.,
“Mult
ip
oin
t
and
Mu
lti-O
b
jectiv
e
A
e
ro
dynam
ic
Sha
p
e
O
ptim
izat
ion,”
AI
AA
J
ournal
,
V
o
l.
42
,
No.
6
,
200
4,
pp
.
10
57–1
065.
27
T
ruon
g,
A.
H.,
O
ldfi
eld,
C.,
an
d
Zin
gg,
D.
W.,
“A
Line
ar
Elastic
it
y
Mes
h
Mo
v
em
en
t
Meth
o
d
with
an
A
u
gme
n
t
ed
Adjoin
t
Appro
ac
h
for
Aero
dynam
ic
Shap
e
Op
timi
zatio
n,”
P
ro
cee
ding
s
of
the
12th
Ann
ua
l
C
A
S
I
Aero
dynam
ics
Symp
osiu
m,
T
o
ron
to,
pa
p
e
r
317
,
200
7.
28
T
ruon
g,
A
.
H.,
Old
field
,
C
.,
a
nd
Zin
gg,
D.
W.,
“Mesh
Mo
v
em
en
t
for
a
Disc
rete-Ad
j
oi
n
t
New
ton-
Krylo
v
Algo
rithm
for
Aero
dyna
mic
O
pti
miza
tion,

AIA
A,
2007
-395
2,
20
07.
29
Ho
w
e
,
M.
S
.,
“Th
e
i
nflue
nce
of
su
rf
ac
e
ro
und
ing
on
traili
ng
edge
noi
se,”
Jo
u
r
n
a
l
of
S
o
u
nd
and
Vibr
a
tion
,
V
o
l.
1
26,
No.
3,
1988
,
pp.
503
–523
.
30
Rump
f
k
eil,
M.
and
Z
ingg
,
D.
W.
,
“A
Genera
l
F
ra
mew
o
rk
for
t
he
O
ptim
al
Co
n
trol
of
Unst
eady
Flo
w
s
with
Applic
atio
ns,”
AI
AA,
2
007
-1128
,
20
07.
31
Rump
f
k
eil,
M.
and
Zi
ngg,
D.
W.,
“O
ptim
al
Aeroac
ousti
c
Shap
e
Des
ign
Using
a
Di
screte
Adjoin
t
App
roac
h,”
Pro
ceed
ings
of
th
e
15t
h
Ann
ua
l
Co
nf
e
renc
e
of
th
e
CFD
So
ciet
y
of
Ca
nada
,
T
oron
to
,
pa
p
e
r
110
6,
pp
.
22
1-22
8,
20
07.
32
Rump
f
k
eil,
M.
an
d
Zin
gg,
D.
W.
,
“Th
e
R
e
mote
I
n
v
er
se
Sha
p
e
Design
of
Airfoi
ls
in
Unste
ady
Flo
ws,”
P
ro
ce
edin
gs
of
the
12
th
Ann
ua
l
CASI
Aero
dyn
amic
s
Symp
osi
um,
T
oron
to,
pap
er
3
18,
2007
.
9
o
f
9
Ame
rican
I
n
stitu
te
of
Aero
naut
ics
a
nd
Astr
onau
tics