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Scientific Programmes Committee
Centre for Aerospace Systems Design & Engineering
K. SudhakarDepartment of Aerospace Engineering
Indian Institute of Technology, Mumbai
http://www.casde.iitb.ac.in/MDO/3d-duct/
July 5, 2003
3D-Duct Design
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Design Optimization / MDO
So far . . .
Airborne Early Warning System (M Tech)
Complex system, simple models.
Maneuver Load Control (M Tech)
Existing system, database driven Hypersonic Launch Vehicle (Ph D)
New system, simple models, system analysis
WingOpt Wing Design (4 x M Tech)
Simple models
Intermediate level models
FEM + VLM
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3D-Duct Optimization
Joint exercise - CASDE + ADA
First attempt at CFD based optimization
Literature
Techniques to inject CFD into optimization
About the design Problem
Capturing of design problem
Parametrization Capturing designers thumb rules & heuristics to trim
design space.
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Optimization
n
k
m
n
SdesignsFeasible
g0)x(g
h0)x(h
;toSubjectx)x(fMinimise
=
=
How to reduce CFD analysis requirements? If gradient based optimization is used; how to
evaluate derivatives?
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Gradient Based
X1
X2
Gradient of
functions
Required!
=
n
2
1
x
f
x
f
xf
)x(f
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3-D Duct DesignDesign Problem in Brief
Entry ExitLocation and shape known
Geometry of duct from Entry to Exit ?
PressureRecovery?
Distortion? Swirl?
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Parametrization of 3D-Ducts
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3D-Duct Design UsingHigh Fidelity Analysis
?
X1-MIN X1-MAX
X2-MAX
X2-MIN
Domain for search using high fidelity code is large
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3D-Duct Design UsingHigh Fidelity Analysis
Low Fidelity Design Criteria Wall angle < 6
Diffusion angle < 3
6 * REQ < ROC
Fluent for CFD
RSM / DOE
DACEX1-MIN X1-MAX
X2-MAX
X2-MIN
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Surrogate Modeling
DOE / RSM modeling in physical experiments.
experimental point
RSM. Least Square Fit.
y = a0 + a1 x + a2 x2 . . .
Fitted model is smooth and easily differentiable.
Curse of dimensionality! 2k function evaluations
Sequential RSM.
x
y
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Sequential RSM
Reported @ICIWIM
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Design & Analysis of Computer Experiments
Regression fit + Stochastic process Single global fit
Variability in prediction known and exploitable
xx
xx
x
Estimates of
Predictive error
x = Computer exp
DACE Fit
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Building Models Using DACE
xx
xx
x
5% predictive
error
x = Computer exp
DACE Fit
xx
x
Use multi-modal GA to identify n highest peaks.
Test if they are higher than 5%
Add computer experiments at those spots
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Homotopy / Continuation
If you seek f(x) = 0
Create a parametric problem;
g(x, ) = ( 1 - ) h(x) + f(x)
Solution to h(x) = 0 is known;ie. g(x,0) = 0 is known
Vary slowly from 0 to 1
g(x, 1) = 0 = f(x)
Solution for duct-1 ( = 0) is known
Solve for duct-2 ( = 1) by slowly varying
= 0
= 1
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How to evaluate gradients?
Consider design of wings;
Design variables, x = [x1, x2]
Objective function, f(x)
Analysis is CFD Give values to x = [x1, x2] duct mesh Run a CFD code and generate solution
Generate f(x) based on solution.
How to evaluate
?2x
f;?
1x
f=
=
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Methods to Evaluate Gradients?
Finite difference method. Easy to implement, but
problematic?
Complex variables approach, requires source
ADIFOR Automatic DIfferentation in FORtran;requires source. Analytical accuracy
Surrogate Modeling Surface fits
Response Surface Method (RSM / DOE) Design & Analysis of Computer Experiments
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Problem with Finite Differencing?
Only (n+1) CFD runs?
Correct step size for FDM is important!
Will demand more CFD runs!
b
CLIterative
Convergence
Criteria
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Complex Variable Approach
Evaluate f{x + i e} ; e
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Gradients by ADIFOR
Complex Analysis
Code in FORTARN
FORTRAN
source code
that can evaluate
gradients
Automated
Differentiation
Package
Euler code is
being put through
ADIFOR
(Not for 3D-Duct)