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SAROD 2003 1
Aerodynamic Design Optimization Studies at CASDE
Amitay Isaacs, D Ghate, A G Marathe,
Nikhil Nigam, Vijay Mali,
K Sudhakar, P M Mujumdar
Centre for Aerospace Systems Design and Engineering
Department of Aerospace Engineering, IIT Bombay
http://www.casde.iitb.ac.in
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SAROD 2003 2
About CASDE
5 years old Master’s program in Systems Design & Engineering MDO MAV Modeling & Simulation Workshops/CEPs/Conferences
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SAROD 2003 3
Optimization Studies –Overview
Concurrent aerodynamic shape & structural sizing of wing FEM based aeroelastic design MDO architectures WingOpt software
Propulsion system Engine sizing & cycle design Intake duct design using CFD
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SAROD 2003 4
Intake Design - Background
Duct design practice of late 80s – based on empirical rules
Problem Revisited – using formal optimization and high fidelity analysis
Study evolved with active participation of ADA (Dr. T.G. Pai & R.K.Jolly)
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SAROD 2003 5
Problem Formulation
Entry Exit Location and shape (Given)
Optimum geometry of duct from Entry to Exit ?
Objective/Constraints
• Pressure Recovery• Distortion• Swirl
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SAROD 2003 6
Design Using CFD - Issues
Simulation Time CFD takes huge amounts of time for real life problems Design requires repetitive runs of disciplinary
analyses
Integration & Automation Parametric geometry modeling Grid generation CFD solution Objective/Constraint function evaluation Optimization
Gradient Information Finite difference – step size (??), (NDV + 1) analyses
required Exact formulations – Automatic differentiation
(ADIFOR), Adjoint method, Complex step method – All require source code
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SAROD 2003 7
Flow Solver
Distortion & Swirl calculation requires NS solutionIn-house NS Solver
Analytical gradients possible Easy to integrate
Commercial Solvers (STAR-CD, FLUENT…) Gradients using finite difference only Difficult to integrate
FLUENT Inc. S-shaped non-diffusing duct Results validated with a NASA test case
(Devaki Ravi Kumar & Sujata Bandyopadhyay)
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SAROD 2003 8
Strategies
Reducing Time Parameterization Variable fidelity to shrink the search space Surrogate modeling Meshing Parallel computing Continuation
Integration & Automation Wrapping executables and user interfaces Offline analysis (Surrogate models) – semi-
automatic
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SAROD 2003 9
Our Strategy
Variable fidelity Response Surface based design using FLUENT
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SAROD 2003 10
Our Methodology
Parametrization
Low fidelity Analysis
DOE in reduced space
CFD analysis at DOE points
RS for PR & DC60
OptimizationConstraint
s
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SAROD 2003 11
Parametrization
Y
X
Z
XDuct Centerline
A
X
Control / Design Variables
• Ym, Zm
• AL/3, A2L/3
Cross Sectional Area
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SAROD 2003 12
Y
X
Z
XDuct Centerline
A
X
Control / Design Variables
• Ym, Zm
• AL/3, A2L/3
Cross Sectional Area
Parametrization
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SAROD 2003 13
Typical 3D-Ducts
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SAROD 2003 14
Duct Design - Low Fidelity
Low Fidelity Design Rules (Constraints) Wall angle < 6° Diffusion angle < 3° 6 * Equivalent Radius
< ROC of Centerline
Objective function: pressure recovery
No low fidelity analysis for distortion or swirl
X1-MIN
X2-MIN
X2-MAX
X1-MAX
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SAROD 2003 15
Optimization Process – Low Fidelity
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SAROD 2003 16
Automation for CFD
Generation of entry and exit sections using GAMBIT
Clustering Parameters
Conversion of file format to CGNS using FLUENT
Mesh file
Generation of structured volume grid using parametrization
Duct Parameters(β1, β2, αy, αz)
Entry & Exit sections
Conversion of structured grid to unstructured format
Unstructured CGNS file
CFD Solution using FLUENT
End-to-end (Parameters to DC60)
automated CFD Cycle. Objective/Constraints evaluationUsing UDFs (FLUENT)
DC60
CFD Solution
ContinuationSolution
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SAROD 2003 17
Automation for Design
Generation of structured volume grid using parametrizationEntry & Exit
sections
Conversion of structured grid to unstructured format
CFD Solution using FLUENT
Objective/Constraints evaluationUsing UDFs (FLUENT)
DC60
Optimization
Duct Parameters(β1, β2, αy, αz)
ContinuationSolution
Unstructured CGNS file
CFD Solution
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SAROD 2003 18
Results: Total Pressure Profile
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SAROD 2003 19
Design Space Reduction
6.19
1.42
(0.61, 0.31, 1.0, 1.0)
Optimized duct from low fidelity
24.2116.28DC60
3.532.0PLOSS
(-0.4, 1.5, 0.3, 0.6)
(0.1, 0.31, 0.2, 0.6)
P
Poor ductInfeasible duct
P – Parameters; PLOSS – Total Pressure Loss
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SAROD 2003 20
Optimization Post-processing
Distortion Analysis DC60 = (PA0 – P60min) /q
where, PA0 - average total pressure at the section,
P60min- minimum total pressure in a 600 sector, q - dynamic pressure at the cross section.User Defined Functions (UDF) and scheme files were used to generate this information from the FLUENT case and data file.Iterations may be stopped when the distortion values stabilize at the exit section with reasonable convergence levels.
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SAROD 2003 21
Huge benefits as compared to the efforts involved!!!
Methodology Store the solution in
case & data files Open the new case (new grid)
with the old data file Setup the problem Solution of (0.61 0.31 1 1) duct slapped on (0.1 0.31 0.1 0.1)
3-decade-fall 6-decade-fall
Without continuation 4996 9462
With continuation 1493 6588
Percentage time saving 70% 30%
Continuation Method
Generate new case file
FLUENT Solution
Duct Parameters
OldData
file
Journal
file
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SAROD 2003 22
Simulation Time
Strategies Continuation Method Parallel execution of FLUENT on a 4-
noded Linux cluster
Time for simulation has been reduced to around 20%.
0 20 40 60 80 100
Time (hrs)
Time per CFD Run
Serial
Parallel
Slapping
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SAROD 2003 23
Sequential (Multipoint)Response Surface Approximations
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SAROD 2003 24
Sequential (multipoint) Response Surface Methodology
Response Surfaces generated in sub-domains around multiple pointsSurfaces used to march to optimum
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SAROD 2003 25
Wing aerodynamic design problem
Planform fixed2 spanwise stations4 variables for camber3 variables for geometric pre-twistMaximize cruise L/D Lift constraint
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SAROD 2003 26
Design Problem Statement
Maximize L/D Sub. to CL = .312
-5 r + m 5 -5 r + m + t 5
with side constraints, .05 x1 .33; .001 h1 .1
.05 x2 .33; .001 h2 .1
-2 r 5 -2 m 5 -2 t 5
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SAROD 2003 27
Design Tools
Lift Calculation: CL from VLM
Drag Calculation: CD0 from a/c data
CDi from VLM
DOE: Design Expert D-optimality Criterion
Response Surfaces: Design Expert quadratic/cubic
Optimizers : FFSQP
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SAROD 2003 28
Overall Design Procedure
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SAROD 2003 29
Results - Arbitrary Starting Point 1
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SAROD 2003 30
Results - Arbitrary Starting Point 2
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SAROD 2003 31
Observations
Quadratic model found better than cubic model in subspaces.
Global model inadequate.
Cost of D-optimality significant
SRSA seems to work well!
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SAROD 2003 32
GRADIENT INFORMATION BY
AUTOMATIC DIFFERENTIATION OF
CFD CODES
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SAROD 2003 33
User Supplied Analytical Gradients
AnalysisCode in Fortran
Manually extractsequence of mathematical
operations
Code the complex derivative evaluator
in Fortran
Manually differentiatemathematical
functions - chain rule
FORTRANsource code
that can evaluategradients
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SAROD 2003 34
Automatic Differentiation for Analytical Gradients
Automatically parse and extract the sequence
of mathematical operations
Use symbolic math packages to automate derivative evaluation
Automatically code the complex
derivative evaluator in Fortran
AnalysisCode in FORTARN
FORTRANsource code
that can evaluategradients
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SAROD 2003 35
Automatic Differentiation for Analytical Gradients
Complex AnalysisCode in FORTARN
FORTRANsource code
that can evaluategradients
Automated Differentiation
Packageeg. ADIFOR
&ADIC
Euler
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SAROD 2003 36
1.12 3.06 4.11
d(L/D) / d using ADIFOR 5.48 -0.38 -1.20
d(L/D) / d using Finite Difference
=0.2
Value 5.09 -0.52 -1.23
% Error 7.17 38.10 2.46
=0.02Value 5.44 -0.40 -1.18
% Error 0.70 4.44 1.73
=0.002Value 5.45 -0.41 -1.18
% Error 0.61 7.08 1.56
=0.0002
Value 5.56 -0.67 -1.02
% Error 1.54 77.25 15.09
Comparison of Derivative Calculation Finite Difference vs ADIFOR
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SAROD 2003 37
Optimization - ADIFOR vs FD
Single design variable unconstrained optimization problem Find for max. L/D for Onera M6 wing
Same starting point; FD step size 0.002
init optL/Dopt Calls Time
(min.)
ADIFOR
1.060 2.810 11.99 15 424
FD 1.060 2.810 11.99 17 111
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SAROD 2003 38
Thank You
Please visitwww.casde.iitb.ac.in
for details and other information
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Thank You
http://www.casde.iitb.ac.in/mdo/3d-duct/
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SAROD 2003 40
Problem Statement
•Ambient conditions: 11Km altitude• Inlet Boundary Conditions
• Total Pressure: 34500 Pa• Total Temperature: 261.4o K• Hydraulic Diameter: 0.394m• Turbulence Intensity: 5%
• Outlet Boundary Conditions• Static Pressure: 31051 Pa (Calculated for the desired mass flow rate)• Hydraulic Diameter: 0.4702m• Turbulence Intensity: 5%
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SAROD 2003 41
Duct Parameterization
Geometry of the duct is derived from the Mean Flow Line (MFL) MFL is the line joining centroids of
cross-sections along the duct Any cross-section along length of the
duct is normal to MFL
Cross-section area is varied parametrically Cross-section shape in merging area is same as the exit section
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SAROD 2003 42
MFL Design Variables - 1Mean flow line (MFL) is considered as a piecewise cubic curve along the length of the duct between the entry section and merging section
x
y(x), z(x)
0 LmLm/2
y(Lm/2), z(Lm/2) specified
Centry
Cmerger
y1, z1
y2, z2
Lm : x-distance between the entry and merger section
y1, y2, z1, z2 : cubic polynomials for y(x) and z(x)
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SAROD 2003 43
MFL Design Variables - 2
• y1(x) = A0 + A1x + A2x2 + A3x3, y2(x) = B0 + B1x + B2x2 + B3x3
• z1(x) = C0 + C1x + C2x2 + C3x3, z2(x) = D0 + D1x + D2x2 + D3x3
• y1(Lm) = y2 (Lm), y1’ (Lm) = y2’ (Lm), y1” (Lm) = y2” (Lm)
• z1(Lm) = z2 (Lm), z1’ (Lm) = z2’ (Lm), z1” (Lm) = z2” (Lm)
• y1’ (Centry) = y2’ (Cmerger) = z1’ (Centry) = z2’ (Cmerger) = 0
• The shape of the MFL is controlled by 2 parameters which control the y and z coordinate of centroid at Lm/2
• y(Lm/2) = y(0) + (y(L) – y(0)) αy 0 < αy < 1
• z(Lm/2) = z(0) + (z(L) – z(0)) αz 0 < αz < 1
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SAROD 2003 44
Area Design Variables – 1Cross-section area at any station is interpolated from the entry and exit cross-sections
•A(x) = A(0) + (A(Lm) – A(0)) * β(x)
• corresponding points on entry and exit sections are linearly interpolated to obtain the shape of the intermediate sections and scaled appropriately
• Psection = Pentry + (Pexit - Pentry) * β
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SAROD 2003 45
Area Design Variables - 2
A0 + A1x + A2x2 + A3x3 0 β < β1
B0 + B1x + B2x2 + B3x3 β1 β β2
C0 + C1x + C2x2 + C3x3 β2< β 1
β =
x
β(x)
0 LmLm/30
1
2Lm/3
β1
β2
β(Lm/3) and β(2Lm/3) is specified
β variation is given by piecewise cubic curve as function of x
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SAROD 2003 46
Turbulence ModelingRelevance: Time per SolutionFollowing aspects of the flow were of interest:
Boundary layer development Flow Separation (if any) Turbulence Development
Literature Survey S-shaped duct Circular cross-section Doyle Knight, Smith, Harloff, Loeffer
Baldwin-Lomax model (Algebraic model) Computationally inexpensive than more sophisticated models Known to give non-accurate results for boundary layer separation etc.
Devaki Ravi Kumar & Sujata Bandyopadhyay (FLUENT Inc.) k- realizable turbulence model
Two equation model
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SAROD 2003 47
Turbulence Modeling (contd.)
Standard k- model Turbulence Viscosity Ratio
exceeding 1,00,000 in 2/3 cells
Realizable k- model Shih et. al. (1994) Cμ is not assumed to be
constant A formulation suggested
for calculating values of C1 & Cμ
Computationally little more expensive than the standard k- model
Total Pressure profile at the exit section (Standard k- model)
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SAROD 2003 48
Results
Mass imbalance: 0.17%Energy imbalance: 0.06%Total pressure drop: 1.42%Various turbulence related quantities of interest at entry and exit sections:
Entry Exit
Turbulent Kinetic Energy (m2/s2)
124.24 45.65
Turbulent Viscosity Ratio 5201.54 3288.45
y+ at the cell center of the cells adjacent to boundary throughout the domain is around 18.
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SAROD 2003 49
Flow Separation
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SAROD 2003 50
Flow Separation