Adjoint-based Trailing-Edge Noise Minimization via Porous Material Beckett Y. Zhou 1,2 , Nicolas R. Gauger 1 , Seong R. Koh 3 , Matthias Meinke 3 , Wolfgang Schr¨ oder 3 1 Chair for Scientific Computing, TU Kaiserslautern, Germany 2 Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Germany 3 Institute of Aerodynamics (AIA), RWTH Aachen University, Germany 19th Euro AD Workshop, Kaiserslautern April 7, 2016
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Adjoint-based Trailing-Edge Noise Minimization viaPorous Material
Beckett Y. Zhou1,2, Nicolas R. Gauger1,
Seong R. Koh3, Matthias Meinke3, Wolfgang Schroder3
1Chair for Scientific Computing, TU Kaiserslautern, Germany2Aachen Institute for Advanced Study in Computational Engineering Science (AICES),
Mean flow/source quantities ( · )o based on an offline sampling stage
0
0 0
Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 19/ 24
Coupled CFD-CAA Framework
~x =
εKv
Kt
LES APE
Mean Flow Field~Qo = [ρo , ~uo , po ]T
Mean Source Term~Lo = (~ω × ~u)o
J =√
(p′)2ρ, ~u, p ρ′, ~u ′, p′
6
Full chain algorithmically differentiated for dJd~x
Mean source term (~Lo) and flow field ( ~Qo) recomputed from free-streamafter each design update
Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 20/ 24
Coupled CFD-CAA Framework
~x =
εKv
Kt
LES APE
Mean Flow Field~Qo = [ρo , ~uo , po ]T
Mean Source Term~Lo = (~ω × ~u)o
J =√
(p′)2ρ, ~u, p ρ′, ~u ′, p′
6
Full chain algorithmically differentiated for dJd~x
Mean source term (~Lo) and flow field ( ~Qo) recomputed from free-streamafter each design update
Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 20/ 24
Coupled CFD-CAA Framework
~x =
εKv
Kt
LES APE
Mean Flow Field~Qo = [ρo , ~uo , po ]T
Mean Source Term~Lo = (~ω × ~u)o
J =√
(p′)2ρ, ~u, p ρ′, ~u ′, p′
6
Full chain algorithmically differentiated for dJd~x
Mean source term (~Lo) and flow field ( ~Qo) recomputed from free-streamafter each design update
Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 20/ 24
Acoustic Pressure in Near-Far Field
35 40 45 50 55 60tUo/h
p′/ρoa2 o
Acoustic Pressure at 3 Observer Locations
BaselineOptimized
2 × 10−3
Optimization Window
Obs. Loc. #1
Obs. Loc. #2
Obs. Loc. #3
x/c
y/c
0.8 0.85 0.9 0.95 1 1.050.05
0
0.05
0.1
0.15
c=33h
h
0.06c 0.06c
1 2 3
d = 0.12c
Observer NoiseLocation Reduction (dB)
1 15.52 17.53 14.2
RMS of p′ reducedby 87% over 3observer locations
Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 21/ 24
Conclusion and Future Work
ConclusionFirst work in noise-minimization to couple discrete adjoint with ahigh-fidelity LES and LES-APE solver
Significant noise reduction achieved by minimizing the pressurefluctuation at off-body observation points – 12dB in the normaldirection and up to 18dB in the upstream directions.
Preliminary result on coupled LES-APE solver also shows effectivesuppression of acoustic pressure fluctuation in the near far-field observerpoints via optimal distribution of porous material in the trailing edge
Take-away MessagesAdjoint-based method allows for exploration of large design spacesNon-intuitive designs possible without unnecessarily penalizing otherperformance metricsAlgorithmic differentiation leads to accurate & stable adjoint informationover long integration times – particularly well-suited for design problemsin aeroacoustics
Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 22/ 24
Conclusion and Future Work
ConclusionFirst work in noise-minimization to couple discrete adjoint with ahigh-fidelity LES and LES-APE solver
Significant noise reduction achieved by minimizing the pressurefluctuation at off-body observation points – 12dB in the normaldirection and up to 18dB in the upstream directions.
Preliminary result on coupled LES-APE solver also shows effectivesuppression of acoustic pressure fluctuation in the near far-field observerpoints via optimal distribution of porous material in the trailing edge
Take-away MessagesAdjoint-based method allows for exploration of large design spacesNon-intuitive designs possible without unnecessarily penalizing otherperformance metricsAlgorithmic differentiation leads to accurate & stable adjoint informationover long integration times – particularly well-suited for design problemsin aeroacoustics
Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 22/ 24
Conclusion and Future Work
Future WorkPerform optimizations on a full 3D turbulent case at increased spanwiseresolution
Allow variations of porosity/permeability parameters in the spanwisedirection(Stay tuned and join us at the 22nd AIAA Aeroacoustics Conference in Lyon,
France: Session AA-14, May 30)
Apply methodology to aerodynamic shapes – airfoil or wing with poroustrailing edge
Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 23/ 24
Acknowledgements
Financial support from German Research Foundation (DFG) andCanadian Postgraduate Scholarship (NSERC-PGS-D)
Computing resources provided by the “Alliance of High PerformanceComputing Rheinland-Pfalz” (AHRP), via the “Elwetritsch” Cluster atthe TU Kaiserslautern
Thank you for your attention
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Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 24/ 24
Pressure Fluctuation Field
Baseline Optimized
Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 24/ 24
AD-based Unsteady Discrete Adjoint Framework
Consider a system of semi-discretized PDEs as follows:
dU
dt+ R(U) = 0
U: spatially discretized state vectorR(U): is the discrete spatial residual vector.
Second-order backward difference is used for time discretization:
R∗(Un) =3
2∆tUn + R(Un)− 2
∆tUn−1 +
1
2∆tUn−2 = 0, n = 1, . . . ,N
Dual-time stepping method converges R∗(Un) to a steady state solution at eachtime level n through a pseudo time τ :
dUn
dτ+ R∗(Un) = 0
Implicit Euler method is used to time march the above equation to steady state:
Unp+1 − Un
p + ∆τR∗(Unp+1) = 0, p = 1, . . . ,M
Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 24/ 24
AD-based Unsteady Discrete Adjoint Framework
The resultant nonlinear system can be linearized around Unp to solve for the state
Unp+1:
Unp+1 − Un
p + ∆τ
[R∗(Un
p ) +∂R∗
∂U
∣∣∣∣np
(Unp+1 − Un
p )
]= 0, p = 1, . . . ,M
This can be written in the form of a fixed-point iteration:
Unp+1 = G n(Un
p ,Un−1,Un−2), p = 1, . . . ,M, n = 1, . . . ,N
G n: an iteration of the pseudo time steppingUn−1: converged state vector at time level n − 1Un−2: converged state vectors at time level n − 2
The fixed point iteration converges to the numerical solution Un:
Un = G n(Un,Un−1,Un−2), n = 1, . . . ,N
Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 24/ 24
AD-based Unsteady Discrete Adjoint Framework
The discretized unsteady optimization problem over N time levels:
minβ
J =1
N
N∑n=1
J(Un, β)
subject to Un = G n(Un,Un−1,Un−2, β), n = 1, . . . ,N
β: vector of design variables. One can express the Lagrangian associated with theabove constrained optimization problem as follows:
L =1
N
N∑n=1
J(Un, β)−N∑
n=1
[(Un)T (Un − G n(Un,Un−1,Un−2, β)
)]Un: adjoint state vector at time level n.
∂L
∂Un= 0, n = 1, . . . ,N (State equations)
KKT :∂L
∂Un= 0, n = 1, . . . ,N (Adjoint equations)
∂L
∂β= 0, (Control equation)
Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 24/ 24
AD-based Unsteady Discrete Adjoint Framework
The unsteady discrete adjoint equations can be derived in the fixed point form as:
Uni+1 =
(∂G n
∂Un
)T
Uni +
(∂G n+1
∂Un
)T
Un+1 +
(∂G n+2
∂Un
)T
Un+2 +1
N
(∂Jn
∂Un
)T
, n = N, . . . , 1
Un+1: converged adjoint state vector at time level n + 1Un+2: converged adjoint state vector at time level n + 2
The unsteady adjoint equations above are solved backward in time.The sensitivity gradient can be computed from the adjoint solutions:
dL
dβ=
N∑n=1
(1
N
∂Jn
∂β+ (Un)T
∂G n
∂β
)High-lighted terms computed using AD in reverse mode
Reverse accumulation used at each time level to ‘tape’ the computationalgraph for AD
Adjoint iterator inherits the same convergence properties as primal iterator
G includes: turbulence model, grid movement, limiters, etc
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Run-time and Memory Usage
3.5 million cellsPrimal solution: 10GB, constant at all time stepsReverse-mode AD: 42GB per time step, scales with number of time stepsSlow-down factor: ∼15 (primal vs. black-box reverse mode AD)
Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 24/ 24
Minimization of Trailing Edge Turbulence Intensity
Strip Viscous Permeabilities
1 2 3 4 50
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Strip Number
Vis
co
us P
erm
eab
ilit
y (
Kv)
baseline
optimized
Optimizer makes the laststrip almost an impermeablesolid (constraint active)
Unclear a priori why thispermeability distribution isoptimal
Highlights the power ofcombining high-fidelitysimulation with numericaloptimization – opportunity toexplore non-intuitive andunconventional designs
Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 24/ 24