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Parameter Sensitivity Analysis for Hypersonic ViscousFlow using
a Discrete Adjoint Approach
Brian A. Lockwood and Dimitri J. Mavriplis
Mechanical EngineeringUniversity of Wyoming
January 4, 2010
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 1 / 32
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Outline
Outline
Introduction
Overview of Flow Solver and Physical Models
Solution Scheme Details
Sensitivity Formulation
Flow and Sensitivity Results for Perfect Gas Model
Flow and Sensitivity Results for Real Gas Model
Conclusion
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 2 / 32
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Introduction
Simulation of complexproblems rely heavily onempirical
relations
Relations can requirehundreds of parameters todefine
Sensitivity and uncertaintycan enhance analysis anddesign
capability
Sampling currently used dueto nonlinear nature of flows
Thousands of flowsolutions requiredComputationallyexpensive
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 3 / 32
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Introduction
Localized sensitivity calculated usingflow adjoint
Sensitivity to large number of inputswith cost approximately
equal to flowsolve
Possibilities for uncertaintyquantification, adaptation
andsimulation optimization.
Should be possible to augmentweaknesses of sampling with an
adjointbased approach
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 4 / 32
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Flow Solver Details
Navier Stokes Equations:
∂U
∂t+∇ · ~F (U) = ∇ · ~Fv (U) + S(U) (1)
Two dimensional, cell-centered finite volume solver using
unstructuredtriangles and/or quadrilaterals
Solver uses a fully implicit, pseudo-time stepping method
Perfect gas and Real gas models examined
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 5 / 32
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Physical Model
Perfect gas variables
U =
ρρ~uρet
~F =
ρ~uρ~u ⊗ ~u + P
ρ~uht
~Fv =
0τ
τ · ~u − ~q
Sutherland’s Law used for viscosity
5 Parameters required to define model:
Ratio of Specific Heats γReynolds NumberPrandtl NumberTwo
constants within Sutherland’s Law
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 6 / 32
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Physical Model
5 species, Two Temperature Real Gas Model
U =
ρsρ~uρetρev
~F =
ρs~uρ~u ⊗ ~u + P
ρ~uhtρ~uhv
S =
ωs00∑
s ωsD̂v ,s + QT−V
~Fv =
−ρs Ṽsτ
τ · ~u − ~q − ~qv −∑
s ht,sρs Ṽs−∑
s hv ,sρs Ṽs − ~qv
Dunn-Kang chemical kinetics model used
Transport quantities calculated using curve fits from Blottner
et al
Approximately 250 constants required to define physical
model
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 7 / 32
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Solver Description
Solution marched to steady state using implicit pseudo-time
stepping
Method of Lines:dU
dt+ R(U) = 0 (2)
BDF1 discretization used for pseudo-time derivative
Unsteady residual given by:
J(Un,Un−1) =Un −Un−1
∆t+ R(Un) (3)
Nonlinear equation J(Un,Un−1) = 0 solved approximately at
eachtime step
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 8 / 32
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Solution Scheme
Fixed number of inexact Newton iterations performed per time
step(∼ 10)
δUk = − [P]−1 J(Uk ,Un−1) (4)Uk+1 = Uk + λδUk (5)
[P] chosen to approximate ∂J∂Uk
, 1st order Van-Leer-Hänel
Preconditioner matrix and transport quantities calculated once
pertime step and frozen.
Global time stepping used for start-up, local time stepping for
fullconvergence
λ used to keep updates sensible
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 9 / 32
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Spatial Discretization
Gradient reconstruction of primitives
Green-Gauss contour integration used to calculate gradients
Smooth Van Albada Limiter with Pressure Switch used:
Ψk = max(0, 1− Kνk)1
∆−(∆+
2+ ε2)∆− + 2∆−
2∆+
∆+2 + 2∆− + ∆−∆+ + ε2(6)
νi =
∑k |PR − PL|∑k PR + PL
(7)
Face based Gradients calculated using averaging and correction
term:
∇Vk = ∇̃V +VR − VL − ∇̃V ·∆~T
|∆~T |∆~T
|∆~T |(8)
Inviscid Flux Calculated Using AUSM+UP flux function with
FrozenSpeed of Sound
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 10 / 32
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Sensitivity Derivation
Let the objective and constraint have following functional
dependence
L = L(D,U(D)) (9)
R = R(D,U(D)) = 0 (10)
Objective and Constraint may be differentiated using the Chain
rule
dL
dD=∂L
∂D+∂L
∂U
∂U
∂D(11)
dR
dD=∂R
∂D+∂R
∂U
∂U
∂D= 0 (12)
Solve Constraint Equation for ∂U∂D
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 11 / 32
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Sensitivity Derivation
Forward Sensitivity Equation Given by:
dL
dD=∂L
∂D− ∂L∂U
∂R
∂U
−1 ∂R
∂D(13)
Transpose Equation
dL
dD
T
=∂L
∂D
T
− ∂R∂D
T ∂R
∂U
−T ∂L
∂U
T
(14)
Define Flow Adjoint with Equation:
∂R
∂U
T
Λ = − ∂L∂U
T
(15)
Sensitivity Calculated by:
dL
dD
T
=∂L
∂D
T
+∂R
∂D
T
Λ (16)
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 12 / 32
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Parameter Sensitivity
For Model Parameters, D dependence enters through field
variables µ
L = L(U(D),µ(D,U(D))) (17)
R = R(U(D),µ(D,U(D))) (18)
Sutherland’s Law Example
µ
µref=
C1T3/2
T + S(19)
Forward Sensitivity Equation for Model Parameters
dL
dD=∂L
∂µ
∂µ
∂D+
(∂L
∂U+∂L
∂µ
∂µ
∂U
)∂U
∂D(20)
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 13 / 32
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Parameter Sensitivity
Adjoint Sensitivity Equation for Model Parameters
∂L
∂D
T
=∂µ
∂D
T(∂L
∂µ
T
+∂R
∂µ
T
Λ
)(21)
Flow Adjoint Found by Solving Equation:[∂R
∂U+∂R
∂µ
∂µ
∂U
]TΛ = −
(∂L
∂U
T
+∂µ
∂U
T ∂L
∂µ
T)
(22)
Adjoint Equation solved with Defect Correction:
[P]T δΛk = − ∂L∂U
T
−
[∂µ
∂U
T ∂R
∂µ
T
+∂R
∂U
T]
Λk (23)
Λk+1 = Λk + δΛk (24)
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 14 / 32
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Field Variable Sensitivity
Set ∂µ∂D = 1 and∂µ∂U = 0:
∂L
∂µ
T
=∂L
∂µ
T
+∂R
∂µ
T
Λµ (25)
Formally, Adjoint Solution with Frozen µ required:[∂R
∂U
)µ
]TΛµ = −
∂L
∂U
)Tµ
(26)
Field Variables defined throughout domain (at cell centers or
facecenters)
May be used for Uncertainty Propagation or Model Adaptation
δL2 =∑
i
∂L
∂µi
2
δµ2i (Ui ) (27)
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 15 / 32
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Perfect Gas Results
5 km/s cylinder test case
Fixed Wall temperature
Results compared againstLAURA and FUN2D
Table: Benchmark Flow Conditions
V∞ = 5 km/sρ∞ = 0.001 kg/m
3
T∞ = 200 KTwall = 500 KM∞ = 17.605Re∞ = 753,860Pr∞ = 0.72
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 16 / 32
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Perfect Gas Results
Temperature Contour for 5 km/s case
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 17 / 32
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Perfect Gas Results
Temperature along Stagnation Streamline Temperature along
Stagnation Streamline(Boundary Layer)
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 18 / 32
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Perfect Gas Results
Surface Pressure Distribution Surface Heating Distribution
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 19 / 32
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Perfect Gas Sensitivity
Geometric and Parameter Sensitivities for 5 km/s Benchmark
CaseSensitivities computed for Integrated Surface heating
Objective:
L =
∫A−k∇T · ~ndA (28)
Sensitivities Validated using Finite DifferenceAdjoint Solution
for Velocity
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 20 / 32
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Geometric Sensitivity
Sensitivity of Surface Heating to surface geometry
calculated
Normal displacement of surface grid points used as design
variable
31 total design variables
-50 0 50Angle (degrees)
-2e+06
0
2e+06
4e+06
Sen
siti
vity
Adjoint SensitivityFinite Difference Result
Heat Flux Sensitivity to Surface Point Displacements
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 21 / 32
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Parameter Sensitivity
Sensitivity to Perfect Gas Parameter γ
Fixed Freestream Velocity and Fixed Freestream Mach
NumberConsidered
Table: Sensitivity of Surface Heating to γ
Fixed Velocity Fixed Mach Number
Objective Value 1.628× 10−2 1.628× 10−2Finite Difference −3.111×
10−2 −1.198× 10−2
Adjoint −3.109× 10−2 −1.206× 10−2Relative Error 7.279× 10−4
7.186× 10−4
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 22 / 32
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Real Gas Results
5 km/s cylinder test case
Fixed Wall temperature
Super-catalytic Wall
Results compared withLAURA
Table: Benchmark Flow Conditions
V∞ = 5 km/sρ∞ = 0.001 kg/m
3
T∞ = 200 KTwall = 500 KM∞ = 17.605Re∞ = 753,860Pr∞ = 0.72
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 23 / 32
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Real Gas Results
Temperature along Stagnation Streamline Temperature along
Stagnation Streamline (Log Scale)
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 24 / 32
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Real Gas Results
Mass Fraction along Stagnation Streamline for 5 km/s case
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 25 / 32
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Real Gas Results
Skin Friction Distribution Surface Heating Distribution
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 26 / 32
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Real Gas Sensitivity
Sensitivities computed for Integrated Surface heating
Objective:
L = −∫
Ak∇T · ~n + kv∇Tv · ~ndA (29)
Sensitivities Validated using Finite Difference
Geometric Sensitivity, 61 Design Variables
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 27 / 32
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Real Gas Sensitivity
Sensitivity of Surface Heating to Arrhenius Coefficients:
Kf = Cf Tηfa e−
Ea,fkTa (30)
Kb = CbTηba e−
Ea,bkTa (31)
Sensitivities Expressed as fractional changes (i.e. dL/LdD/D
)
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 28 / 32
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Real Gas Sensitivity
Sensitivity of Surface Heating to Species Viscosity Curve
FitParameters:
µs = 0.1e(As ln(T )+Bs)ln(T )+Cs (32)
Significantly Higher Sensitivity than Reaction Parameters
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 29 / 32
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Real Gas Sensitivity
Sensitivity of Surface Heating to Face-centered Viscosity
throughoutDomain7800 Faces within Computational GridExact and
Approximate Adjoints used to Compute Sensitivity
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 30 / 32
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Real Gas Sensitivity
Viscosity Sensitivity along Stagnation Streamline Viscosity
Sensitivity at 45 degrees
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 31 / 32
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Conclusions
Conclusions:
Sensitivity to large number of parameters with minimal effort
possiblewith adjoint
Parameter sensitivity can be used to determine most important
modelcomponents
Field variable sensitivity possible; identifying regions/ranges
ofgreatest importance
Future Work:
Extend valid range of sensitivities by including higher order
terms
Investigate simulation adaptation using adjoint
Explore hybrid sensitivity/sampling approaches to
uncertaintyquantification
B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint
Sensitivity Jan. 4, 2010 32 / 32