Transcript Multi-disciplinary Design Optimization

3D-Duct Design
Scientific Programmes Committee
Centre for Aerospace Systems Design & Engineering
K. Sudhakar
Department of Aerospace Engineering
Indian Institute of Technology, Mumbai
http://www.casde.iitb.ac.in/MDO/3d-duct/
July 5, 2003
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
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.
Optimization
Minimise
f (x)
x  n
h (x)  0
h  m
g (x)  0
g  k
Subject to;
Feasible designs  S   n
• How to reduce CFD analysis requirements?
• If gradient based optimization is used; how to
evaluate derivatives?
Gradient Based
Gradient of functions
Required!
X2



X1
 f 
 x 
 f1 


x 2 

f (x) 






 f 
 x n 
3-D Duct Design
Design Problem in Brief
• Pressure Recovery?
• Distortion?
• Swirl?
Entry
Exit
Location and shape known
Geometry of duct from Entry to Exit ?
Parametrization of 3D-Ducts
3D-Duct Design Using
High Fidelity Analysis
X2-MAX
?
X2-MIN
X1-MIN
X1-MAX
Domain for search using high fidelity code is large
3D-Duct Design Using
High Fidelity Analysis

X2-MAX


X2-MIN

X1-MIN
X1-MAX
Low Fidelity Design Criteria
 Wall angle < 6°
 Diffusion angle < 3°
 6 * REQ < ROC
Fluent for CFD
RSM / DOE
DACE
Surrogate Modeling
DOE / RSM modeling in physical experiments.
experimental point
y
RSM. Least Square Fit.
y = a0 + a 1 x + a2 x 2 . . .
Fitted model is smooth and easily differentiable.
Curse of dimensionality! 2k function evaluations
Sequential RSM.
x
Sequential RSM
Reported @ICIWIM
Design & Analysis of Computer Experiments
• Regression fit + Stochastic process
• Single global fit
• Variability in prediction known and exploitable
x = Computer exp
DACE Fit
x
x
x
x
x
Estimates of
Predictive error
Building Models Using DACE
x
x
x
x
x
x x
x x = Computer exp
DACE Fit
5% predictive
error
Use multi-modal GA to identify ‘n’ highest peaks.
Test if they are higher than 5%
Add computer experiments at those spots
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
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
f
f
 ?;
?
 x1
x 2
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
Problem with Finite Differencing?
Only (n+1) CFD runs?
Iterative
Convergence
Criteria
CL
b
Correct step size for FDM is important!
Will demand more CFD runs!
Complex Variable Approach
subroutine func (x, f)
real x, f
subroutine func(x, f)
complex x, f
Evaluate f{x + i e} ; e << 1
f(x) = Real Part { f(x + i e) }
- f”(x) e2 / 2
df/dx = Imag Part { f(x+ i e) } / e - f ”’(x) e2 / 6
CPU time up by 3, RAM up by 2
Gradients by ADIFOR
Complex Analysis
Code in FORTARN
Euler code is
being put through
ADIFOR
(Not for 3D-Duct)
FORTRAN
source code
that can evaluate
gradients
Automated
Differentiation
Package