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National Aeronautics and Space Administration
Coupled CFD/Sonic Boom Adjoint
Methodology and its Application to Aircraft
Design
Sriram Rallabhandi National Institute of Aerospace
CFD Seminar
1
Outline
2
1. Introduction and motivation
2. Advanced Boom Analysis
3. Sonic boom adjoint methodology and derivation
4. Verification of adjoint sensitivities
5. Coupled adjoint formulation and derivation
6. Results
7. Discussion, Conclusions and Future work
3
Introduction and objectives
Primary Objectives
• Establish capability to design and analyze supersonic vehicles that
achieve efficiency improvement & sonic boom reduction.
• Demonstrate ability to design and optimize for improved cruise
efficiency or sonic boom reduction for both under-track and off-track
• Sonic boom is the intense noise produced by aircraft flying at
supersonic speeds
• One of supersonic flight’s biggest challenges remains
mitigation of sonic boom
Sonic Boom
CFD off-body dp/p
Geometry
Motivation and Goals: BIG PICTURE
CFD Flow Solver
CFD Adjoint Solver
Ground Signature
Target Ground Signature or Target Loudness
sBOOM
sBOOMAdjoint
Current Design
Practice
Target Near-field
Advanced Boom Analysis – Primal Problem
Pc
PG
PCPP
PP
cG
00
2
2
221
1 002
2
• Boom propagation based on augmented Burgers’ equation
• Solution process involves operator splitting scheme
• Non-linearity solved using: ),(),( PPP
• Crank-Nicolson finite difference scheme used for absorption
and relaxation
)(
33
22
1
nn
n
n
n
n
n
n
n
n
n
n
nn
n
tfp
rBtA
qBrA
pBkqA
• Above equations repeated from the off-body location to the
ground
• Boom signature obtained after ground reflection
• Numerically equivalent to solving the following set of equations
Advanced Boom Analysis – Primal Problem
• Geometrical spreading:
sB
OO
M
),()(
)(),( P
G
GP
• Atmospheric stratification: ),()]([
)]([),( P
c
cP
00
00
sBOOM, 90.7 PLdB
PCBoom w/ tanh 86.9 PLdB
PCBoom
• Lossy propagation code predicts shock
rise times to properly compute the
frequency spectrum and hence loudness
of the ground signatures
Lossy Propagation
Loss-less Propagation
Advanced Boom Analysis – Primal Problem
• sBOOM captures the frequency
and magnitude of the ground
signatures using physics-based
algorithms
• Compares well with a similar code
8
Shape Optimization: Needs and Goals
• Key ingredients for effective high-fidelity shape optimization for
sonic boom reduction
• Repeatable, reliable CFD and CFD adjoint solutions [FUN3D]
• Usable geometry parameterization [MASSOUD/BANDAIDS]
• Using ground signatures or metrics to drive the CFD adjoints
• Good optimization formulation and problem setup
• Primary goals:
• Formulation and derivation of the boom adjoint problem
• Prediction and verification adjoint sensitivities of a ground
based cost function w.r.t. selected design variables
• Boom Adjoint + CFD coupling for shape optimization to
reduce boom impact
• Demonstrate adjoint shape optimization through effective
design cases [email protected]
9
Boom Adjoint Derivation
• Discrete-adjoint Lagrangian
1
1
11
1
10
1
33
1
22
1
11
2
0
1
)()()(
)()(),(
,
,,
DBkqAtfprBtA
qBrApBkqADplL
T
nnn
N
n
T
nnn
n
n
nN
n
T
n
nn
n
n
nN
n
T
nnn
n
nn
nN
n
T
n
N
n
nn
• Derivative of the Lagrangian
1
1
111
1,0
1
33
1
22
1
,1
1
2
,0
1
)( BkD
qA
D
t
t
f
D
p
D
rB
D
tA
D
qB
D
rA
D
pBk
D
qAl
D
p
p
l
D
l
dD
dL
T
nn
n
nN
n
T
n
nnnnn
N
n
T
nnnnnn
N
n
T
n
nnn
nnn
N
n
T
n
N
n
nnn
n
nn
10
Boom Adjoint Equations
• Sonic boom discrete-adjoint equations:
• Cost/Objective functions on the ground:
nT
n
nT
n
nT
n
nT
n
n
nT
n
nT
n
n
n
T
n
n
nT
n
BA
BA
t
fA
Bkp
l
210
321
3
1
110
,,
,
,
itiN
N
N
M
i
itinn
ppp
l
ppl
,,
,, )(
01
2
21
, if n=N
, if n≠N
• Gradient of the objective from Lagrangian:
1
1
11,0 BkdD
dL T
N
t
N
N
tN
p
dBAdBAdBA
p
l
dBAdBAl
)(2
2
sB
OO
MA
djo
int
11
Verification of Adjoint Sensitivities - Loudness
• Adjoint sensitivities are verified against those from complex
variable approach
Axial Location (ft)
p0
=d
p/p
dL
/dp
0
450 500 550 600 650 700 750 800-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
-2000
-1000
0
1000
2000Initial Pressure Waveform
dL/dp0
Time (ms)
dp
(ps
f)
d(d
BA
)/d
PN
0 50 100 150-0.4
-0.2
0
0.2
-30
-20
-10
0
10
20
30
Ground Signature
d(dBA)/dPN
12
Verification of Adjoint Sensitivities - Target
• Adjoint sensitivities computed
• Sensitivities verified against those
from complex variable approach
• Sample target chosen
13
Coupled-Adjoint Formulation
0
0
00
b
T
f
T
g
T
b
T
f
T
b
T
N
X
T
X
R
X
G
Q
T
Q
R
dp
dl
)(),,,,,( TpRGlXQDLT
b
T
f
T
gNbgf 0
• Discrete coupled-adjoint Lagrangian
• CFD coupled sonic boom discrete-adjoint equations:
• Derivative of the ground objective w.r.t. aircraft shape
parameters
D
R
D
G
dD
dL T
f
T
g
14
Verification of Coupled-Adjoint Sensitivities
• Boom adjoint coupled with FUN3D Adjoint
• Baseline parameterized (Massoud/Bandaids) for shape
optimization
• Derivatives from sBoom adjoint + FUN3D adjoint compared
against Complex sBoom + Complex FUN3D
• Good match (up to 10th decimal digit)
15
Results – Target Matching
• Design ground signature closer to target than baseline signature
• PLdB reduction from 83.7 to 83.5 (far from the target loudness of
79.0) • Objective reduces by slightly > 50%
Time (ms)
p(p
sf)
0 50 100-0.4
-0.2
0
0.2
Baseline (PLdB = 83.7)
Optimum (PLdB = 83.5)
Target (PLdB = 79)
Design Cycle
Ob
jec
tiv
e
5 10 15 20
0.15
0.2
0.25
0.3
Results – Loudness Minimization
Design Cycle
Ob
jec
tiv
e
5 10 15 20 2540
45
50
55
60
65
70
75
80
85
90
Design Cycle
Ob
jec
tiv
e
5 10 15 20 25 30
30
35
40
45
50
55
60
• Overall, good PLdB reduction from 83.7 to 80.6
• More importantly, shaped front and aft signatures
• L/D reduces from 8.38 to 7.88
Time (ms)
p(p
sf)
0 50 100-0.4
-0.2
0
0.2
0.4
Baseline (69.5 dBA, 83.7 PLdB, L/D=8.38)
Aft part (66.6 dBA, 81.6 PLdB, L/D=8.08)
Wing + Aft part (65.5 dBA, 80.6 PLdB,L/D=7.88)
Results – CFD Integration
• The off-body pressure
distribution is plotted here
• The shape changes (Blue and
Red) and plotted on top of the
baseline (grey) – some non-
intuitive changes to the
geometry Axial Location (ft)
dp
/p
500 550 600 650-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
Baseline
Aft part
Wing + Aft part
Discussion, Conclusions, and Future Work
• A discrete adjoint boom propagation method based on
augmented Burgers’ equation has been developed
• Integration with CFD off-body adjoint design capability is a
powerful tool
• CFD-coupled boom adjoint allows use of efficient gradient
based optimization techniques in supersonic aircraft design
• Test cases demonstrate the potential of the method – however a
lot needs to be done w.r.t parameterization and optimizer setup
• Ideas for future work:
• Better integration and demonstration with adjoint CFD including using
engine simulation, and viscous (N-S) solutions
• Multi-point, multi-objective robust design studies
• Obtain sensitivity with respect to other propagation parameters (Relaxation,
relative humidity etc. etc.)
