Post on 25-Jul-2020
1/19
Application of Adjoint Methodology in Various
Aspects of Sonic Boom Design
Sriram K. Rallabhandi, National Institute of Aerospace
In support of NASA High Speed Project
• AIAA Aviation 2014
• Sonic Boom Activities III
• June 17, 2014
2/19
Outline
• Current status of adjoint shape optimization applied to
sonic boom mitigation
• Goals
• Extension to boom adjoint theory
• Verification of adjoint sensitivities
• Results
• Discussion
• Summary
Sriram.Rallabhandi@nasa.gov
3/19
Geometry
Current Status: Adjoint-Based Shape Optimization for Sonic Boom Mitigation
CFD off-body dp/p
Target Near-field
Design
Analysis
4/19
Geometry
Current Status: Adjoint-Based Shape Optimization for Sonic Boom Mitigation
Ground metric
Off-body interface
Analysis
Design
sBOOM
5/19
Geometry
Current Status: Adjoint-Based Shape Optimization for Sonic Boom Mitigation
Reversed Equivalent
area Location
Reversed Equivalent
area distribution
Target
hr
Off-body interface
Analysis
Design
6/19
Goals
Extend sonic boom propagation utility (sBOOM1,2) to:
Generate target equivalent areas using adjoint
sensitivities
─ Targets at multiple azimuths in the
neighborhood of the baseline design
─ Efficient compared to non-gradient based
optimization approaches
Obtain sensitivity of sonic boom metrics to:
─ Flight conditions
─ Propagation parameters
─ Atmospheric quantities
Sriram.Rallabhandi@nasa.gov
1Rallabhandi, S. K., “Advanced Sonic-Boom Prediction Using the Augmented Burgers Equation”, Journal of Aircraft, Vol. 48, pp: 1245-1253, 2011 2Rallabhandi, S. K., Nielsen, E. J., Diskin, B., “Sonic-Boom Mitigation Through Aircraft Design and Adjoint Methodology”, Journal of Aircraft, Vol. 51, pp: 502-510, 2014
7/19
Discrete Boom Adjoint Formulation
• Discrete-adjoint Lagrangian
)()()(
)()(),(
1
11
1
1,0
1
33
1
22
1
,11
2
,0
1
DBkqAtfprBtA
qBrApBkqADpJL
T
nn
N
n
T
nn
n
n
nN
n
T
n
n
n
n
nN
n
T
nn
n
nn
nN
n
T
n
N
n
n
• Derivative of the Lagrangian
)( 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
qA
D
p
p
J
D
J
dD
dL
Tn
n
nN
n
T
n
nnnnN
n
T
nnnnn
N
n
T
n
nn
nnn
N
n
T
n
N
n
n
n
nn
𝐷 = 𝑃0
𝑘𝑛 − Blokhintzev scaling term 𝐴𝑛, 𝐵𝑛 − 𝑁2 Relaxation matrices 𝐴2
𝑛, 𝐵2𝑛 − 𝑂2 Relaxation matrices
𝐴3𝑛, 𝐵3
𝑛 − Absorption matrices 𝑓 𝑡𝑛 − Nonlinear terms
)(
33
22
1
nn
n
n
n
n
n
n
n
n
n
n
nn
n
tfp
rBtA
qBrA
pBkqA
sBO
OM
8/19
Boom Adjoint Equations: Existing Theory
• Sonic boom discrete-adjoint equations:
• Gradient of the objective:
1
11,0 BkdD
dL T
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
J
2,1,0
32,1
3
1
11,0
sBO
OM
Ad
join
t
)(
33
22
1
nn
n
n
n
n
n
n
n
n
n
n
nn
n
tfp
rBtA
qBrA
pBkqA
sBO
OM
00
2
)(2
p
dBAdBAdBA
p
J
dBAdBAJ
t
t
o
etee
o
T
teetee
p
AAA
p
J
AAAAJ
,
,,21 ))((
• Cost functions:
9/19
Boom Adjoints: Extension to Existing Theory
• Change of independent variable vector from p0 to Ae
y
xy
AdxyF
edAdF
r
M
pddL
edA
pd
pddL
edAdL
e
0
21
0
0
0
2/1
''
)(
2
2
)(
)(
)(
X (ft)
Ae
(sq
.ft
)
0 50 100 1500
10
20
30
40
Baseline Target
Optimized Target
2 4 6 8 100
0.2
0.4
14 15 16 17 18
1
1.2
1.4
Time (ms)
p
(ps
f)
0 20 40 60 80 100 120-0.2
-0.1
0
0.1
0.2
0.3 Baseline, PLdB = 68.24
Optimized, PLdB = 66.27
X
F-f
un
cti
on
0 50 100 150-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Baseline F-function
Optimized F-function
• Gradient optimization leads to numerically
better, but practically worse targets
Existing formulation
1xN 1xN NxN 1xN 1x1 NxN
Numerical differentiation
10/19
Boom Adjoints: Extension to Existing Theory
• Change of independent variable vector from p0 to Ae
y
xy
AdxyF
edAdF
r
M
pddL
edA
pd
pddL
edAdL
e
0
21
0
0
0
2/1
''
)(
2
2
)(
)(
)(
Existing formulation
1xN 1xN NxN 1xN 1x1 NxN
• Smoothing is needed
Cubic spline based Ae targets based on control points
Algorithm not only returns spline interpolation, but also the
Jacobian matrices:𝜕𝐴𝑒
𝜕𝑋𝐶𝑃,
𝜕𝐴𝑒
𝜕𝐴𝑒,𝐶𝑃 [N > C, CP = Control Points]
Gradient computation extended by chain rule
NxC NxC
1xC 1xN NxC
Numerical differentiation
11/19
Sensitivities wrt flight conditions and propagation parameters
D
t
t
f
D
p
D
Br
D
rB
D
At
D
tA
D
Bq
D
qB
D
Ar
D
rA
D
kpB
D
Bpk
D
pBk
D
Aq
D
qA
D
p
p
J
D
J
dD
dL
n
n
nN
n
T
n
nT
nnn
nT
nnn
N
n
T
n
nT
nnn
nT
nnn
N
n
T
n
nn
nn
T
nnnn
n
nT
nnn
N
n
T
n
N
n
n
n
nn
1
33
33
1
22
22
1
,1
111
1
,0
1
• Independent variable vector D:[∆𝜎, 𝑆, Γ, 𝜃𝜐,1, 𝐶𝜐,1, 𝜃𝜐,2, 𝐶𝜐,2 , ℎ, 𝑀]
• Updated derivative of the Lagrangian
D
Br
D
At
D
Bq
D
Ar
D
kpB
D
Bpk
D
Aq
dD
dL
nT
n
nT
n
N
n
T
n
nT
n
nT
n
N
n
T
n
nn
nn
T
nn
nT
n
N
n
T
n
33
1
22
1
,1
11
1
,0
• Updated gradient calculation:
• Significant increase in memory requirement (~12 GB more than
previous formulation for a typical case)
Previous gradient computation:𝑑𝐿
𝑑𝐷= 𝛾0,1
𝑇 𝑘1𝐵1
1xM 1xM 1xN NxM NxN NxM NxNxM 1xN 1x1 NxN NxM 1xN 1x1 NxNxM
NxNxM NxNxM NxNxM NxNxM NxN NxN NxN NxN
1xN
1xN 1xN NxM NxM NxM NxM 1xN 1xN 1xN 1xN
1xN Nx1 NxN 1xM
NxM NxM NxN
12/19
Verification of Adjoint Sensitivities
Sriram.Rallabhandi@nasa.gov
• Adjoint sensitivities verified against complex step gradients
Imaginary step size of 10-50 used in evaluating complex gradients
Good match up to 8 digits of numerical accuracy for target Ae
• Sensitivities wrt flight conditions, and propagation parameters
13/19
Results: Target Ae Generation
• 15 spline control points
• SQP optimization with A-
weighted loudness (dBA) to
be minimized
• 64-bit Xeon CPU with
16GB memory
• Total wall time ~ 30
minutes
• Under- and off-track targets
generated simultaneously
• Targets generated in the
neighborhood of the
baseline Sriram.Rallabhandi@nasa.gov
14/19
Results: Target Ae generation
• Typically convergence achieved in 50-60 function evaluations
• A-weighted loudness (dBA) minimized, but perceived
loudness tracked
Good correlation between dBA and PLdB
Sriram.Rallabhandi@nasa.gov
15/19
Results: Atmospheric Sensitivity
Sriram.Rallabhandi@nasa.gov
• Loudness sensitivity to Mach number decreases with Mach
number
• Loudness sensitivity to cruise altitude approaches zero in
tropopause
16/19
Results: Atmospheric Sensitivity
Sriram.Rallabhandi@nasa.gov
• Loudness is very sensitive to step size at extremely low
step sizes
Quickly approaches zero for higher numbers
• Loudness sensitivity to number of points used during
propagation asymptotically approaches zero
17/19
Discussion
• Adjoint-based design optimization used to generate targets
Targets generated in literature were based on either non-gradient
approaches or linearized boom minimization theory
New approach provides an efficient way to generate targets
• Adjoint-based atmospheric sensitivity analysis
Some sources of epistemic uncertainties considered
Possible extension to include aleatory uncertainties such as
temperature, winds, and relative humidity profiles
Quantify uncertainty during sonic boom propagation
Sriram.Rallabhandi@nasa.gov
Robust design point where error and sensitivity are simultaneously
minimized
Use information to understand and improve design
18/19
Summary
• sBOOM framework extended to generate boom
sensitivities with respect to equivalent areas, flight
conditions, and propagation parameters
• Target equivalent areas generated using gradient-based
optimization at multiple azimuthal angles
• Sensitivity of boom metrics to flight conditions and
propagation parameters obtained, plotted and
observations made
Sriram.Rallabhandi@nasa.gov
19/19
QUESTIONS? Acknowledgments:
• NASA High Speed project
• Jim Fenbert for setting up the ModelCenter model
for gradient-based optimization to generate target
equivalent areas
• Karl Geiselhart (ModelCenter)
• Irian Ordaz (off-track analysis)
• Mathias Wintzer for manuscript review and general
discussions
• Lori Ozoroski for manuscript review and support