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The Cost Of Not Using
Active Subspaces
Uncertainty Quantification of the HyShot II Scramjet
Michael Emory | Paul Constanti...
Chapter 5
Active Subspaces in
Action
arXiv:submit/1052518
SIAM CSE 3-14-2015 Salt Lake City, UT 1
Outline
Sensitivity Analysis & R.O.M.
Gradient-based Response Surface
Active Subspace
comparison of three approaches:
SIAM...
Outline
Sensitivity Analysis & R.O.M.
Gradient-based Response Surface
Active Subspace
HyShot II Scramjet
Uncertainties
SIA...
Outline
Sensitivity Analysis & R.O.M.
Gradient-based Response Surface
Active Subspace
HyShot II Scramjet
Uncertainties
SIA...
SIAM CSE 3-14-2015 Salt Lake City, UT 5
NASA X-43A
miles per hour
7000
What is a scramjet?
HyShot II Scramjet Geometry
SIAM CSE 3-14-2015 Salt Lake City, UT 6
forebody ramp inlet/isolator combustor nozzle/afterbod...
HyShot II Scramjet Geometry
SIAM CSE 3-14-2015 Salt Lake City, UT 7
forebody ramp inlet/isolator combustor nozzle/afterbod...
HyShot II Scramjet Geometry
SIAM CSE 3-14-2015 Salt Lake City, UT 8
forebody ramp inlet/isolator combustor nozzle/afterbod...
HyShot II Failure: Unstart
forebody ramp inlet/isolator combustor nozzle/afterbody
numerical Schlieren
SIAM CSE 3-14-2015 ...
HyShot II Simulation
SIAM CSE 3-14-2015 Salt Lake City, UT 10
forebody ramp inlet/isolator combustor nozzle/afterbody
nume...
HyShot II Simulation
SIAM CSE 3-14-2015 Salt Lake City, UT 11
forebody ramp inlet/isolator combustor nozzle/afterbody
nume...
HyShot II Scramjet Simulations
SIAM CSE 3-14-2015 Salt Lake City, UT 12
forebody ramp inlet/isolator combustor nozzle/afte...
Sensitivity Analysis & R.O.M.
Gradient-based Response Surface
Active Subspace
HyShot II Scramjet
Uncertainties
SIAM CSE 3-...
Sources of Uncertainty
SIAM CSE 3-14-2015 Salt Lake City, UT 14
α vehicle angle of attack
I turbulence intensity
Lt turbul...
Sources of Uncertainty
SIAM CSE 3-14-2015 Salt Lake City, UT 15
α vehicle angle of attack
I turbulence intensity
Lt turbul...
Sources of Uncertainty
SIAM CSE 3-14-2015 Salt Lake City, UT 16
α vehicle angle of attack
I turbulence intensity
Lt turbul...
Sources of Uncertainty
SIAM CSE 3-14-2015 Salt Lake City, UT 17
α vehicle angle of attack
I turbulence intensity
Lt turbul...
Sensitivity Analysis & R.O.M.
Gradient-based Response Surface
Active Subspace
HyShot II Scramjet
Uncertainties
SIAM CSE 3-...
Sensitivity Analysis & R.O.M.
Gradient-based Response Surface
Active Subspace
HyShot II Scramjet
Uncertainties
SIAM CSE 3-...
Plan of Action
SIAM CSE 3-14-2015 Salt Lake City, UT 20
Sensitivity Analysis dimension reduction
Tensor Grid Sampling gath...
1D Sensitivity Study
SIAM CSE 3-14-2015 Salt Lake City, UT 21
maxnominalmin
Sensitivity
Sampling
R.O.M.
2D
1D Sensitivity Study
SIAM CSE 3-14-2015 Salt Lake City, UT 22
maxnominalmin
Sensitivity
Sampling
R.O.M.
2D
1D Sensitivity Study
SIAM CSE 3-14-2015 Salt Lake City, UT 23
maxnominalmin
Sensitivity
Sampling
R.O.M.
2D
1D Sensitivity Study
SIAM CSE 3-14-2015 Salt Lake City, UT 24
maxnominalmin
Sensitivity
Sampling
R.O.M.
2D
1D Sensitivity Study
SIAM CSE 3-14-2015 Salt Lake City, UT 25
maxnominalmin
Sensitivity
Sampling
R.O.M.
2D
1D Sensitivity Study
SIAM CSE 3-14-2015 Salt Lake City, UT 26
maxnominalmin
Cost:
15x 2D-sim
Sensitivity
Sampling
R.O.M.
2D
Visual Inspection of Sensitivity
SIAM CSE 3-14-2015 Salt Lake City, UT 27
inflow profile distributions of: pressure
max
no...
Visual Inspection of Sensitivity
SIAM CSE 3-14-2015 Salt Lake City, UT 28
inflow profile distributions of: pressure
max
no...
1000 1500 2000
0.12
0.1225
0.125
0.1275
0.13
1000 1500 2000
0.12
0.1225
0.125
0.1275
0.13
1000 1500 2000
0.12
0.1225
0.125...
Visual Inspection of Sensitivity
SIAM CSE 3-14-2015 Salt Lake City, UT 30
inflow profile distributions of: tke
max
nominal...
Visual Inspection of Sensitivity
SIAM CSE 3-14-2015 Salt Lake City, UT 31
inflow profile distributions of: tke
max
nominal...
Uniform Tensor Sampling
SIAM CSE 3-14-2015 Salt Lake City, UT 32
Sample 5 points per dimension: 55 = 3125 inflow profiles
...
Pressure [Pa]
Y[m]
Uniform Tensor Sampling
SIAM CSE 3-14-2015 Salt Lake City, UT 33
Sample 5 points per dimension: 55 = 31...
tke [m2/s2]Pressure [Pa]
Y[m]
Reduced Order Modeling
SIAM CSE 3-14-2015 Salt Lake City, UT 34
Goal: create 2D surrogate mo...
tke [m2/s2]Pressure [Pa]
Y[m]
Reduced Order Modeling
SIAM CSE 3-14-2015 Salt Lake City, UT 35
Goal: create 2D surrogate mo...
tke [m2/s2]Pressure [Pa]
Y[m]
Reduced Order Modeling
SIAM CSE 3-14-2015 Salt Lake City, UT 36
Goal: create 2D surrogate mo...
Sensitivity Analysis & R.O.M.
Gradient-based Response Surface
Active Subspace
HyShot II Scramjet
Uncertainties
SIAM CSE 3-...
Sensitivity Analysis & R.O.M.
Gradient-based Response Surface
Active Subspace
HyShot II Scramjet
Uncertainties
SIAM CSE 3-...
Response Surfaces
SIAM CSE 3-14-2015 Salt Lake City, UT 39
(actual system behavior)
The basics:
Response Surfaces
SIAM CSE 3-14-2015 Salt Lake City, UT 40
The basics:
requires many evaluations
Response Surfaces
SIAM CSE 3-14-2015 Salt Lake City, UT 41
The basics:
what if I can only afford a
small number of evaluat...
Gradient-enhanced Response Surfaces
SIAM CSE 3-14-2015 Salt Lake City, UT 42
The basics:
additional information
Gradient-enhanced Response Surfaces
SIAM CSE 3-14-2015 Salt Lake City, UT 43
The basics:
additional information
Compute Gradients
SIAM CSE 3-14-2015 Salt Lake City, UT 44
For M=14 samples,
use 1D finite difference to compute gradient ...
Compute Gradients
SIAM CSE 3-14-2015 Salt Lake City, UT 45
For M=14 samples,
use 1D finite difference to compute gradient ...
Compute Gradients
SIAM CSE 3-14-2015 Salt Lake City, UT 46
For M=14 samples,
use 1D finite difference to compute gradient ...
Compute Gradients
SIAM CSE 3-14-2015 Salt Lake City, UT 47
For M=14 samples,
use 1D finite difference to compute gradient ...
Compute Gradients
SIAM CSE 3-14-2015 Salt Lake City, UT 48
For M=14 samples,
use 1D finite difference to compute gradient ...
Compute Gradients
SIAM CSE 3-14-2015 Salt Lake City, UT 49
For M=14 samples,
use 1D finite difference to compute gradient ...
Inflow Profile Insights
SIAM CSE 3-14-2015 Salt Lake City, UT 50
α
I
Lt
P0
H0
xtR
xtC
−0.03 −0.025 −0.02 −0.015 −0.01 −0.0...
Inflow Profile Insights
SIAM CSE 3-14-2015 Salt Lake City, UT 51
α
I
Lt
P0
H0
xtR
xtC
−0.03 −0.025 −0.02 −0.015 −0.01 −0.0...
Inflow Profile Insights
SIAM CSE 3-14-2015 Salt Lake City, UT 52
α
I
Lt
P0
H0
xtR
xtC
−0.03 −0.025 −0.02 −0.015 −0.01 −0.0...
Inflow Profile Insights
SIAM CSE 3-14-2015 Salt Lake City, UT 53
α
I
Lt
P0
H0
xtR
xtC
−0.03 −0.025 −0.02 −0.015 −0.01 −0.0...
Inflow Profile Insights
SIAM CSE 3-14-2015 Salt Lake City, UT 54
−0.2 −0.15 −0.1 −0.05 0 0.05
0.12
0.121
0.122
0.123
0.124...
Compute Gradients
SIAM CSE 3-14-2015 Salt Lake City, UT 55
For M=14 samples,
use 1D finite difference to compute gradient ...
Compute Gradients
SIAM CSE 3-14-2015 Salt Lake City, UT 56
For M=14 samples,
use 1D finite difference to compute gradient ...
Compute Gradients
SIAM CSE 3-14-2015 Salt Lake City, UT 57
For M=14 samples,
use 1D finite difference to compute gradient ...
Understanding Poor Behavior
SIAM CSE 3-14-2015 Salt Lake City, UT 58
Hypothesis:
Finite difference step size is too small,...
Understanding Poor Behavior
SIAM CSE 3-14-2015 Salt Lake City, UT 59
Hypothesis:
Finite difference step size is too small,...
Sensitivity to Step Size
SIAM CSE 3-14-2015 Salt Lake City, UT 60
α
I
Lt
P0
H0
xtR
xtC
−0.03 −0.025 −0.02 −0.015 −0.01 −0....
Sensitivity to Step Size
SIAM CSE 3-14-2015 Salt Lake City, UT 61
α
I
Lt
P0
H0
xtR
xtC
−0.1 −0.05 0 0.05 0.1
0.12
0.121
0....
Sensitivity to Step Size
SIAM CSE 3-14-2015 Salt Lake City, UT 62
α
I
Lt
P0
H0
xtR
xtC
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05
...
Sensitivity to Step Size
SIAM CSE 3-14-2015 Salt Lake City, UT 63
−0.2 −0.15 −0.1 −0.05 0 0.05
0.12
0.121
0.122
0.123
0.12...
Sensitivity to Step Size
SIAM CSE 3-14-2015 Salt Lake City, UT 64
−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1
0...
Sensitivity to Step Size
SIAM CSE 3-14-2015 Salt Lake City, UT 65
−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1
0...
Sensitivity to Step Size
SIAM CSE 3-14-2015 Salt Lake City, UT 66
−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1
0...
Sensitivity to Step Size
SIAM CSE 3-14-2015 Salt Lake City, UT 67
−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1
0...
Sensitivity to Step Size
SIAM CSE 3-14-2015 Salt Lake City, UT 68
−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1
0...
Sensitivity Analysis & R.O.M.
Gradient-based Response Surface
Active Subspace
HyShot II Scramjet
Uncertainties
SIAM CSE 3-...
Discovering the Active Subspace
SIAM CSE 3-14-2015 Salt Lake City, UT 70
Sufficient Summary Plot for M=14 nominal samples
...
Discovering the Active Subspace
SIAM CSE 3-14-2015 Salt Lake City, UT 71
Sufficient Summary Plot for M=14 nominal samples
...
Checking for an Active Subspace
SIAM CSE 3-14-2015 Salt Lake City, UT 72
Sufficient Summary Plot for M=14 nominal samples
...
Checking for an Active Subspace
SIAM CSE 3-14-2015 Salt Lake City, UT 73
Sufficient Summary Plot for M=14 nominal samples
...
Robustness of the Active Subspace
SIAM CSE 3-14-2015 Salt Lake City, UT 74
Sufficient Summary Plot for two independent M=1...
Robustness of the Active Subspace
SIAM CSE 3-14-2015 Salt Lake City, UT
Sufficient Summary Plot for two independent M=14 s...
Robustness of the Active Subspace
SIAM CSE 3-14-2015 Salt Lake City, UT
Sufficient Summary Plot for two independent M=14 s...
Robustness of the Active Subspace
SIAM CSE 3-14-2015 Salt Lake City, UT
Sufficient Summary Plot for two independent M=14 s...
Robustness of the Active Subspace
SIAM CSE 3-14-2015 Salt Lake City, UT
Sufficient Summary Plot for two independent M=14 s...
Summary
SIAM CSE 3-14-2015 Salt Lake City, UT 79
Inflow UQ of the HyShot II scramjet
Summary
SIAM CSE 3-14-2015 Salt Lake City, UT 80
Inflow UQ of the HyShot II scramjet
Cost of not using active subspace
26x...
Summary
SIAM CSE 3-14-2015 Salt Lake City, UT 81
Inflow UQ of the HyShot II scramjet
Cost of not using active subspace
Que...
Auxiliary Slides
SIAM CSE 3-14-2015 Salt Lake City, UT 82
WM-LES Computational Cost
SIAM CSE 3-14-2015 Salt Lake City, UT 83
¼ combustion chamber
FPVA combustion model
coarse mesh:...
HyShot II Scramjet Simulations
SIAM CSE 3-14-2015 Salt Lake City, UT 84
forebody ramp inlet/isolator combustor nozzle/afte...
HyShot II Failure: Unstart
SIAM CSE 3-14-2015 Salt Lake City, UT 85
Goal: characterize safe operating regime
Proxy Indicat...
Visual Inspection of Sensitivity
SIAM CSE 3-14-2015 Salt Lake City, UT 86
inflow profile distributions of: x-velocity
max
...
Visual Inspection of Sensitivity
SIAM CSE 3-14-2015 Salt Lake City, UT 87
inflow profile distributions of: tke
max
nominal...
X-velocity [m/s]
Y[m]
Uniform Tensor Sampling
SIAM CSE 3-14-2015 Salt Lake City, UT 88
Sample 5 points per dimension: 55 =...
tke [m2/s2]
Y[m]
Uniform Tensor Sampling
SIAM CSE 3-14-2015 Salt Lake City, UT 89
Sample 5 points per dimension: 55 = 3125...
Inflow Profile Insights
SIAM CSE 3-14-2015 Salt Lake City, UT 90
α
I
Lt
P0
H0
xtR
xtC
−0.03 −0.025 −0.02 −0.015 −0.01 −0.0...
Inflow Profile Insights
SIAM CSE 3-14-2015 Salt Lake City, UT 91
α
I
Lt
P0
H0
xtR
xtC
−0.03 −0.025 −0.02 −0.015 −0.01 −0.0...
−0.05 0 0.05 0.1 0.15 0.2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Sample#
Consistency of Response
SIAM CSE 3-14-2015 Salt Lake Ci...
−0.05 0 0.05 0.1 0.15 0.2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Sample#
Consistency of Response
SIAM CSE 3-14-2015 Salt Lake Ci...
−0.05 0 0.05 0.1 0.15 0.2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Sample#
Consistency of Response
SIAM CSE 3-14-2015 Salt Lake Ci...
Robustness of the Active Subspace
SIAM CSE 3-14-2015 Salt Lake City, UT
Sufficient Summary Plot for M=14 study
-1 -0.5 0 0...
Robustness of the Active Subspace
SIAM CSE 3-14-2015 Salt Lake City, UT
Weights w:
96
FFR = 25%
FFR = 30%
FFR = 35%
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The Cost of Not Using Active Subspace

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Presentation given at SIAM CSE 2015 on the use of Active Subspaces to quantify uncertainty in the HyShot II scramjet. More importantly this presentation shows a cost/benefit analysis between other methods for parameter reduction. Convincingly demonstrates the simplicity, robustness, and affordability of the active subpace approach.

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The Cost of Not Using Active Subspace

  1. 1. The Cost Of Not Using Active Subspaces Uncertainty Quantification of the HyShot II Scramjet Michael Emory | Paul Constantine | Francisco Palacios | Johan Larsson | Gianluca Iaccarino
  2. 2. Chapter 5 Active Subspaces in Action arXiv:submit/1052518 SIAM CSE 3-14-2015 Salt Lake City, UT 1
  3. 3. Outline Sensitivity Analysis & R.O.M. Gradient-based Response Surface Active Subspace comparison of three approaches: SIAM CSE 3-14-2015 Salt Lake City, UT 2
  4. 4. Outline Sensitivity Analysis & R.O.M. Gradient-based Response Surface Active Subspace HyShot II Scramjet Uncertainties SIAM CSE 3-14-2015 Salt Lake City, UT 3
  5. 5. Outline Sensitivity Analysis & R.O.M. Gradient-based Response Surface Active Subspace HyShot II Scramjet Uncertainties SIAM CSE 3-14-2015 Salt Lake City, UT 4
  6. 6. SIAM CSE 3-14-2015 Salt Lake City, UT 5 NASA X-43A miles per hour 7000 What is a scramjet?
  7. 7. HyShot II Scramjet Geometry SIAM CSE 3-14-2015 Salt Lake City, UT 6 forebody ramp inlet/isolator combustor nozzle/afterbody Flow Mach ~8 numerical Schlieren
  8. 8. HyShot II Scramjet Geometry SIAM CSE 3-14-2015 Salt Lake City, UT 7 forebody ramp inlet/isolator combustor nozzle/afterbody Flow Mach ~8 Shock Train (M ~ 2.5) Bow Shock numerical Schlieren
  9. 9. HyShot II Scramjet Geometry SIAM CSE 3-14-2015 Salt Lake City, UT 8 forebody ramp inlet/isolator combustor nozzle/afterbody Flow Mach ~8 Shock Train (M ~ 2.5) Fuel Injection Mixing and Combustion (M > 1) numerical Schlieren Bow Shock
  10. 10. HyShot II Failure: Unstart forebody ramp inlet/isolator combustor nozzle/afterbody numerical Schlieren SIAM CSE 3-14-2015 Salt Lake City, UT 9 Goal: characterize safe operating regime Proxy QoI: Exit Pressure Fuel Flow Rate
  11. 11. HyShot II Simulation SIAM CSE 3-14-2015 Salt Lake City, UT 10 forebody ramp inlet/isolator combustor nozzle/afterbody numerical Schlieren 2D-sim Domain decomposition: Single physics Span-wise symmetry Cost: 340 core-hours
  12. 12. HyShot II Simulation SIAM CSE 3-14-2015 Salt Lake City, UT 11 forebody ramp inlet/isolator combustor nozzle/afterbody numerical Schlieren 2D-sim Domain decomposition: Single physics Span-wise symmetry Cost: 340 core-hours 3D-sim Multi-physics Fully 3-dimensional Cost: 2600 core-hours
  13. 13. HyShot II Scramjet Simulations SIAM CSE 3-14-2015 Salt Lake City, UT 12 forebody ramp inlet/isolator combustor nozzle/afterbody numerical Schlieren 2D-sim Inflow profile: 300 x 7 matrix Single physics Span-wise symmetry Cost: 340 core-hours 3D-sim Multi-physics Fully 3-dimensional Cost: 2600 core-hours
  14. 14. Sensitivity Analysis & R.O.M. Gradient-based Response Surface Active Subspace HyShot II Scramjet Uncertainties SIAM CSE 3-14-2015 Salt Lake City, UT 13
  15. 15. Sources of Uncertainty SIAM CSE 3-14-2015 Salt Lake City, UT 14 α vehicle angle of attack I turbulence intensity Lt turbulent length scale P0 stagnation pressure H0 stagnation enthalpy xtR ramp transition location xtC cowl transition location Boundary conditions in the 2D-sim: uniform random 7 uncertain parameters:
  16. 16. Sources of Uncertainty SIAM CSE 3-14-2015 Salt Lake City, UT 15 α vehicle angle of attack I turbulence intensity Lt turbulent length scale P0 stagnation pressure H0 stagnation enthalpy xtR ramp transition location xtC cowl transition location Boundary conditions in the 2D-sim: gas state 7 uncertain parameters:
  17. 17. Sources of Uncertainty SIAM CSE 3-14-2015 Salt Lake City, UT 16 α vehicle angle of attack I turbulence intensity Lt turbulent length scale P0 stagnation pressure H0 stagnation enthalpy xtR ramp transition location xtC cowl transition location Boundary conditions in the 2D-sim: turbulence state 7 uncertain parameters:
  18. 18. Sources of Uncertainty SIAM CSE 3-14-2015 Salt Lake City, UT 17 α vehicle angle of attack I turbulence intensity Lt turbulent length scale P0 stagnation pressure H0 stagnation enthalpy xtR ramp transition location xtC cowl transition location Boundary conditions in the 2D-sim: laminar-turbulent transition 7 uncertain parameters:
  19. 19. Sensitivity Analysis & R.O.M. Gradient-based Response Surface Active Subspace HyShot II Scramjet Uncertainties SIAM CSE 3-14-2015 Salt Lake City, UT 18 - d=7 uniform uncertainties - all manifest in 2D domain - QoI requires 2D & 3D simulations - cost of 2D & 3D simulations - inflow profile importance critical knowledge:
  20. 20. Sensitivity Analysis & R.O.M. Gradient-based Response Surface Active Subspace HyShot II Scramjet Uncertainties SIAM CSE 3-14-2015 Salt Lake City, UT 19
  21. 21. Plan of Action SIAM CSE 3-14-2015 Salt Lake City, UT 20 Sensitivity Analysis dimension reduction Tensor Grid Sampling gather function evaluations Reduced Order Modeling dimension reduction 2D simulations Goal: reduce uncertainties from d=7 to d=2
  22. 22. 1D Sensitivity Study SIAM CSE 3-14-2015 Salt Lake City, UT 21 maxnominalmin Sensitivity Sampling R.O.M. 2D
  23. 23. 1D Sensitivity Study SIAM CSE 3-14-2015 Salt Lake City, UT 22 maxnominalmin Sensitivity Sampling R.O.M. 2D
  24. 24. 1D Sensitivity Study SIAM CSE 3-14-2015 Salt Lake City, UT 23 maxnominalmin Sensitivity Sampling R.O.M. 2D
  25. 25. 1D Sensitivity Study SIAM CSE 3-14-2015 Salt Lake City, UT 24 maxnominalmin Sensitivity Sampling R.O.M. 2D
  26. 26. 1D Sensitivity Study SIAM CSE 3-14-2015 Salt Lake City, UT 25 maxnominalmin Sensitivity Sampling R.O.M. 2D
  27. 27. 1D Sensitivity Study SIAM CSE 3-14-2015 Salt Lake City, UT 26 maxnominalmin Cost: 15x 2D-sim Sensitivity Sampling R.O.M. 2D
  28. 28. Visual Inspection of Sensitivity SIAM CSE 3-14-2015 Salt Lake City, UT 27 inflow profile distributions of: pressure max nominal min Sensitivity Sampling R.O.M. 2D Y[m] Y[m] Pressure [Pa] Pressure [Pa]
  29. 29. Visual Inspection of Sensitivity SIAM CSE 3-14-2015 Salt Lake City, UT 28 inflow profile distributions of: pressure max nominal min Sensitivity Sampling R.O.M. 2D
  30. 30. 1000 1500 2000 0.12 0.1225 0.125 0.1275 0.13 1000 1500 2000 0.12 0.1225 0.125 0.1275 0.13 1000 1500 2000 0.12 0.1225 0.125 0.1275 0.13 1000 1500 2000 0.12 0.1225 0.125 0.1275 0.13 1000 1500 2000 0.12 0.1225 0.125 0.1275 0.13 1000 1500 2000 0.12 0.1225 0.125 0.1275 0.13 1000 1500 2000 0.12 0.1225 0.125 0.1275 0.13 Visual Inspection of Sensitivity SIAM CSE 3-14-2015 Salt Lake City, UT 29 inflow profile distributions of: x-velocity max nominal min Sensitivity Sampling R.O.M. 2D
  31. 31. Visual Inspection of Sensitivity SIAM CSE 3-14-2015 Salt Lake City, UT 30 inflow profile distributions of: tke max nominal min Sensitivity Sampling R.O.M. 2D
  32. 32. Visual Inspection of Sensitivity SIAM CSE 3-14-2015 Salt Lake City, UT 31 inflow profile distributions of: tke max nominal min outcome: d=5 Sensitivity Sampling R.O.M. 2D
  33. 33. Uniform Tensor Sampling SIAM CSE 3-14-2015 Salt Lake City, UT 32 Sample 5 points per dimension: 55 = 3125 inflow profiles Sensitivity Sampling R.O.M. 2D
  34. 34. Pressure [Pa] Y[m] Uniform Tensor Sampling SIAM CSE 3-14-2015 Salt Lake City, UT 33 Sample 5 points per dimension: 55 = 3125 inflow profiles Sensitivity Sampling R.O.M. 2D sampled profile mean Cost: 3125x 2D-sim
  35. 35. tke [m2/s2]Pressure [Pa] Y[m] Reduced Order Modeling SIAM CSE 3-14-2015 Salt Lake City, UT 34 Goal: create 2D surrogate model to sample from Sensitivity Sampling R.O.M. 2D X-velocity [m/s] sampled profile mean
  36. 36. tke [m2/s2]Pressure [Pa] Y[m] Reduced Order Modeling SIAM CSE 3-14-2015 Salt Lake City, UT 35 Goal: create 2D surrogate model to sample from Sensitivity Sampling R.O.M. 2D X-velocity [m/s] sampled profile mean Problem: generated inflow profiles aren’t physical
  37. 37. tke [m2/s2]Pressure [Pa] Y[m] Reduced Order Modeling SIAM CSE 3-14-2015 Salt Lake City, UT 36 Goal: create 2D surrogate model to sample from Sensitivity Sampling R.O.M. 2D X-velocity [m/s] sampled profile mean Problem: generated inflow profiles aren’t physical Cost per FFR 15x 2D simulations 3125x 2D simulations R.O.M. expertise associated costs 1.07M core-hours
  38. 38. Sensitivity Analysis & R.O.M. Gradient-based Response Surface Active Subspace HyShot II Scramjet Uncertainties SIAM CSE 3-14-2015 Salt Lake City, UT 37 - maintain d=7 uncertainty space - physical inflow profiles new constraints:
  39. 39. Sensitivity Analysis & R.O.M. Gradient-based Response Surface Active Subspace HyShot II Scramjet Uncertainties SIAM CSE 3-14-2015 Salt Lake City, UT 38
  40. 40. Response Surfaces SIAM CSE 3-14-2015 Salt Lake City, UT 39 (actual system behavior) The basics:
  41. 41. Response Surfaces SIAM CSE 3-14-2015 Salt Lake City, UT 40 The basics: requires many evaluations
  42. 42. Response Surfaces SIAM CSE 3-14-2015 Salt Lake City, UT 41 The basics: what if I can only afford a small number of evaluations?
  43. 43. Gradient-enhanced Response Surfaces SIAM CSE 3-14-2015 Salt Lake City, UT 42 The basics: additional information
  44. 44. Gradient-enhanced Response Surfaces SIAM CSE 3-14-2015 Salt Lake City, UT 43 The basics: additional information
  45. 45. Compute Gradients SIAM CSE 3-14-2015 Salt Lake City, UT 44 For M=14 samples, use 1D finite difference to compute gradient Mi
  46. 46. Compute Gradients SIAM CSE 3-14-2015 Salt Lake City, UT 45 For M=14 samples, use 1D finite difference to compute gradient Mi
  47. 47. Compute Gradients SIAM CSE 3-14-2015 Salt Lake City, UT 46 For M=14 samples, use 1D finite difference to compute gradient Mi
  48. 48. Compute Gradients SIAM CSE 3-14-2015 Salt Lake City, UT 47 For M=14 samples, use 1D finite difference to compute gradient Mi Cost: 112x 2D-sim 112x 3D-sim
  49. 49. Compute Gradients SIAM CSE 3-14-2015 Salt Lake City, UT 48 For M=14 samples, use 1D finite difference to compute gradient Mi Cost: 112x 2D-sim 112x 3D-sim Problem: Response surface doesn’t look good
  50. 50. Compute Gradients SIAM CSE 3-14-2015 Salt Lake City, UT 49 For M=14 samples, use 1D finite difference to compute gradient Mi Cost: 112x 2D-sim 112x 3D-sim Problem: Response surface doesn’t look good Why?
  51. 51. Inflow Profile Insights SIAM CSE 3-14-2015 Salt Lake City, UT 50 α I Lt P0 H0 xtR xtC −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M1
  52. 52. Inflow Profile Insights SIAM CSE 3-14-2015 Salt Lake City, UT 51 α I Lt P0 H0 xtR xtC −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M1
  53. 53. Inflow Profile Insights SIAM CSE 3-14-2015 Salt Lake City, UT 52 α I Lt P0 H0 xtR xtC −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M1
  54. 54. Inflow Profile Insights SIAM CSE 3-14-2015 Salt Lake City, UT 53 α I Lt P0 H0 xtR xtC −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M1
  55. 55. Inflow Profile Insights SIAM CSE 3-14-2015 Salt Lake City, UT 54 −0.2 −0.15 −0.1 −0.05 0 0.05 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M12 α I Lt P0 H0 xtR xtC
  56. 56. Compute Gradients SIAM CSE 3-14-2015 Salt Lake City, UT 55 For M=14 samples, use 1D finite difference to compute gradient Mi Cost: 112x 2D-sim 112x 3D-sim Positive Observations: - direction of response agrees with intuition - magnitude is appropriate
  57. 57. Compute Gradients SIAM CSE 3-14-2015 Salt Lake City, UT 56 For M=14 samples, use 1D finite difference to compute gradient Mi Cost: 112x 2D-sim 112x 3D-sim Positive Observations: - direction of response agrees with intuition - magnitude is appropriate Problem: 6/14 samples display poor behavior
  58. 58. Compute Gradients SIAM CSE 3-14-2015 Salt Lake City, UT 57 For M=14 samples, use 1D finite difference to compute gradient Mi Cost: 112x 2D-sim 112x 3D-sim Positive Observations: - direction of response agrees with intuition - magnitude is appropriate Problem: 6/14 samples display poor behavior Why?
  59. 59. Understanding Poor Behavior SIAM CSE 3-14-2015 Salt Lake City, UT 58 Hypothesis: Finite difference step size is too small, overwhelmed by numerical noise Test: run 2D-sim at
  60. 60. Understanding Poor Behavior SIAM CSE 3-14-2015 Salt Lake City, UT 59 Hypothesis: Finite difference step size is too small, overwhelmed by numerical noise Test: run 2D-sim at Cost: 224x 2D-sim
  61. 61. Sensitivity to Step Size SIAM CSE 3-14-2015 Salt Lake City, UT 60 α I Lt P0 H0 xtR xtC −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M1
  62. 62. Sensitivity to Step Size SIAM CSE 3-14-2015 Salt Lake City, UT 61 α I Lt P0 H0 xtR xtC −0.1 −0.05 0 0.05 0.1 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −1 −0.5 0 0.5 1 1.5 2 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M1
  63. 63. Sensitivity to Step Size SIAM CSE 3-14-2015 Salt Lake City, UT 62 α I Lt P0 H0 xtR xtC −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M1
  64. 64. Sensitivity to Step Size SIAM CSE 3-14-2015 Salt Lake City, UT 63 −0.2 −0.15 −0.1 −0.05 0 0.05 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M12 α I Lt P0 H0 xtR xtC
  65. 65. Sensitivity to Step Size SIAM CSE 3-14-2015 Salt Lake City, UT 64 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.2 −0.15 −0.1 −0.05 0 0.05 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −2 −1.5 −1 −0.5 0 0.5 1 1.5 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M12 α I Lt P0 H0 xtR xtC
  66. 66. Sensitivity to Step Size SIAM CSE 3-14-2015 Salt Lake City, UT 65 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.2 −0.15 −0.1 −0.05 0 0.05 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −4 −3 −2 −1 0 1 2 3 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M12 α I Lt P0 H0 xtR xtC
  67. 67. Sensitivity to Step Size SIAM CSE 3-14-2015 Salt Lake City, UT 66 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.2 −0.15 −0.1 −0.05 0 0.05 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −4 −3 −2 −1 0 1 2 3 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M12 α I Lt P0 H0 xtR xtC Positive Observations: - larger step size improves response - consistent gradient sign at
  68. 68. Sensitivity to Step Size SIAM CSE 3-14-2015 Salt Lake City, UT 67 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.2 −0.15 −0.1 −0.05 0 0.05 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −4 −3 −2 −1 0 1 2 3 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M12 α I Lt P0 H0 xtR xtC Problem: Do you trust the gradient magnitudes?
  69. 69. Sensitivity to Step Size SIAM CSE 3-14-2015 Salt Lake City, UT 68 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.2 −0.15 −0.1 −0.05 0 0.05 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −4 −3 −2 −1 0 1 2 3 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M12 α I Lt P0 H0 xtR xtC Problem: Do you trust the gradient magnitudes? Cost per FFR 336x 2D simulations 112x 3D simulations response surface expertise associated costs 405k core-hours
  70. 70. Sensitivity Analysis & R.O.M. Gradient-based Response Surface Active Subspace HyShot II Scramjet Uncertainties SIAM CSE 3-14-2015 Salt Lake City, UT 69
  71. 71. Discovering the Active Subspace SIAM CSE 3-14-2015 Salt Lake City, UT 70 Sufficient Summary Plot for M=14 nominal samples -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 2.2 2.25 2.3 2.35 2.4 2.45 2.5 active variable weightedexitpressure[bar] FFR = 30%
  72. 72. Discovering the Active Subspace SIAM CSE 3-14-2015 Salt Lake City, UT 71 Sufficient Summary Plot for M=14 nominal samples -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 2.2 2.25 2.3 2.35 2.4 2.45 2.5 active variable weightedexitpressure[bar] FFR = 30% weights w
  73. 73. Checking for an Active Subspace SIAM CSE 3-14-2015 Salt Lake City, UT 72 Sufficient Summary Plot for M=14 nominal samples -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 2.2 2.25 2.3 2.35 2.4 2.45 2.5 active variable weightedexitpressure[bar] FFR = 30% weights w 1D Active Subspace Exists!
  74. 74. Checking for an Active Subspace SIAM CSE 3-14-2015 Salt Lake City, UT 73 Sufficient Summary Plot for M=14 nominal samples -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 2.2 2.25 2.3 2.35 2.4 2.45 2.5 active variable weightedexitpressure[bar] FFR = 30% weights w 1D Active Subspace Exists! But… - are 14 samples really sufficient? - does it exist for other FFRs?
  75. 75. Robustness of the Active Subspace SIAM CSE 3-14-2015 Salt Lake City, UT 74 Sufficient Summary Plot for two independent M=14 studies -1 -0.5 0 0.5 1 1.5 2.2 2.3 2.4 2.5 2.6 2.7 active variable weightedexitpressure[bar] 2013 2015 2015 2013 FFR = 30% weights w
  76. 76. Robustness of the Active Subspace SIAM CSE 3-14-2015 Salt Lake City, UT Sufficient Summary Plot for two independent M=14 studies -1 -0.5 0 0.5 1 1.5 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 active variable weightedexitpressure[bar] 2015 2013 FFR = 35% 2013 2015 75 weights w
  77. 77. Robustness of the Active Subspace SIAM CSE 3-14-2015 Salt Lake City, UT Sufficient Summary Plot for two independent M=14 studies -1 -0.5 0 0.5 1 1.5 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 active variable weightedexitpressure[bar] 2015 2013 FFR = 35% 2013 2015 76 weights w Benefits (if discovered): - robust to noise - robust with very few samples - easy to implement
  78. 78. Robustness of the Active Subspace SIAM CSE 3-14-2015 Salt Lake City, UT Sufficient Summary Plot for two independent M=14 studies -1 -0.5 0 0.5 1 1.5 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 active variable weightedexitpressure[bar] 2015 2013 FFR = 35% 2013 2015 77 weights w Benefits (if discovered): - robust to noise - robust with very few samples - easy to implement M = 2*m; Xhat = 2*rand(m,M)-1; X = 0.5*(repmat(xu-xl,1,M) .* Xhat + repmat(xu+xl,1,M)); for j=1:M f(j) = mymodel(X(:,j)); end Uhat = [ones(M,1) Xhat’] f’; w = uhat(2:m+1)/norm(uhat(2:m+1)); plot(Xhat’*w,f,’o’);
  79. 79. Robustness of the Active Subspace SIAM CSE 3-14-2015 Salt Lake City, UT Sufficient Summary Plot for two independent M=14 studies -1 -0.5 0 0.5 1 1.5 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 active variable weightedexitpressure[bar] 2015 2013 FFR = 35% 2013 2015 78 weights w Benefits (if discovered): - robust to noise - robust with very few samples - easy to implement Cost per FFR 14x 2D simulations 14x 3D simulations basic programming skills 41k core-hours
  80. 80. Summary SIAM CSE 3-14-2015 Salt Lake City, UT 79 Inflow UQ of the HyShot II scramjet
  81. 81. Summary SIAM CSE 3-14-2015 Salt Lake City, UT 80 Inflow UQ of the HyShot II scramjet Cost of not using active subspace 26x sensitivity & R.O.M 10x gradient-based response surface
  82. 82. Summary SIAM CSE 3-14-2015 Salt Lake City, UT 81 Inflow UQ of the HyShot II scramjet Cost of not using active subspace Questions? 26x sensitivity & R.O.M 10x gradient-based response surface
  83. 83. Auxiliary Slides SIAM CSE 3-14-2015 Salt Lake City, UT 82
  84. 84. WM-LES Computational Cost SIAM CSE 3-14-2015 Salt Lake City, UT 83 ¼ combustion chamber FPVA combustion model coarse mesh: 40M Cost: 150k cpu-hours J. Larsson, R. Vicquelin, and I. Bermejo-Moreno, “Large eddy simulation of the HyShot II scramjet”, Center for Turbulence Research Annual Briefs, 2011.
  85. 85. HyShot II Scramjet Simulations SIAM CSE 3-14-2015 Salt Lake City, UT 84 forebody ramp inlet/isolator combustor nozzle/afterbody numerical Schlieren Inflow profile: 300 x 7 matrix pressure contours
  86. 86. HyShot II Failure: Unstart SIAM CSE 3-14-2015 Salt Lake City, UT 85 Goal: characterize safe operating regime Proxy Indicator: Exit Pressure bottomwallpressure x [mm] Increasing Fuel Flow Rate (FFR)
  87. 87. Visual Inspection of Sensitivity SIAM CSE 3-14-2015 Salt Lake City, UT 86 inflow profile distributions of: x-velocity max nominal min Y[m] Y[m] Pressure [Pa] Pressure [Pa]
  88. 88. Visual Inspection of Sensitivity SIAM CSE 3-14-2015 Salt Lake City, UT 87 inflow profile distributions of: tke max nominal min Y[m] Y[m] Pressure [Pa] Pressure [Pa]
  89. 89. X-velocity [m/s] Y[m] Uniform Tensor Sampling SIAM CSE 3-14-2015 Salt Lake City, UT 88 Sample 5 points per dimension: 55 = 3125 inflow profiles Sensitivity Sampling R.O.M 2D sampled profile mean
  90. 90. tke [m2/s2] Y[m] Uniform Tensor Sampling SIAM CSE 3-14-2015 Salt Lake City, UT 89 Sample 5 points per dimension: 55 = 3125 inflow profiles Sensitivity Sampling R.O.M 2D sampled profile mean
  91. 91. Inflow Profile Insights SIAM CSE 3-14-2015 Salt Lake City, UT 90 α I Lt P0 H0 xtR xtC −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M1
  92. 92. Inflow Profile Insights SIAM CSE 3-14-2015 Salt Lake City, UT 91 α I Lt P0 H0 xtR xtC −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.12 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 pressure [Pa] temperature [K] tke [m2/s2]X-velocity [m/s] Δ inflow profile w.r.t. nominalSample M1
  93. 93. −0.05 0 0.05 0.1 0.15 0.2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Sample# Consistency of Response SIAM CSE 3-14-2015 Salt Lake City, UT 92 14 13 12 11 10 9 8 7 6 5 4 3 2 1 zero SampleM
  94. 94. −0.05 0 0.05 0.1 0.15 0.2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Sample# Consistency of Response SIAM CSE 3-14-2015 Salt Lake City, UT 93 14 13 12 11 10 9 8 7 6 5 4 3 2 1 zero SampleM
  95. 95. −0.05 0 0.05 0.1 0.15 0.2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Sample# Consistency of Response SIAM CSE 3-14-2015 Salt Lake City, UT 94 14 13 12 11 10 9 8 7 6 5 4 3 2 1 zero SampleM
  96. 96. Robustness of the Active Subspace SIAM CSE 3-14-2015 Salt Lake City, UT Sufficient Summary Plot for M=14 study -1 -0.5 0 0.5 1 2 2.05 2.1 2.15 2.2 2.25 2.3 active variable weightedexitpressure[bar] FFR = 35% 95 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 2.2 2.25 2.3 2.35 2.4 2.45 2.5 active variable -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 2.45 2.5 2.55 2.6 2.65 2.7 2.75 active variable FFR = 30%FFR = 25%
  97. 97. Robustness of the Active Subspace SIAM CSE 3-14-2015 Salt Lake City, UT Weights w: 96 FFR = 25% FFR = 30% FFR = 35%

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