Optimization.Inverse

OBJ = ∑ (c pdesign − c p actual )2 i i
i =1 N
Starting Point
• The starting point in the design process is a baseline airfoil shape that is known to produce a cp distribution close to the design cp distribution at the design Mach number and angle of attack. • Some expertise is needed to establish a good starting point. • Industries normally start with their best aire called “Bump” Functions
3.5 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 Series1 Series2 Series3 Series4
Design Constraints
• May be geometric:
– Airfoil too thin or too thick.. – Leading edge radius is too small, which may lead to leading edge stall.
• May be performance related:
– Good transonic performance as well as supercruise performance desired. – Good cruise performance as well as high enough Clmax for take-off / landing performance.
Inverse Design Methods
• These methods try to match Cp, actual = Cp, design • They do not attempt to minimize the differences. • Not every Cp design that the designer dreams up will lead to a realistic airfoil!! • We will look at an example approach, called Modified Garabedian McFadden technique.
Some objective functions
• In optimization studies the design objective is usually minimum drag at the design point, or minimum L/D. • However, an accurate prediction of the Cd is currently impossible with most aerodynamic analysis codes for various reasons. • For this reason, many designers prescribe the airfoil pressure distribution cpDesign everywhere on the airfoil and define the objective function to be minimized as
Some Details..
1. The monitor program needs the input baseline geometry, the free-stream Mach number and angle of attack, and the design or target pressures, cpdesign. The main program also reads input regarding which of the 12 design variables are to participate in the design. For example, let a3, a5 and a7 be the three design variables to be considered in the present design. All the other design variables and these three variables are initialized to zero. 2. Optimization routines, say CONMIN, send a3=0, a5=0 and a7=0 through the monitor program to the black box. The values of OBJ are returned. 3. CONMIN finds by perturbing each variable in turn and using finite-difference formulae to evaluate the derivatives as discussed previously. 4. Based on the information from step 3, CONMIN chooses changes to a3, a5 and a7 which will cause OBJ to decrease. These changes are added to a3, a5, a7 to get new updated , etc. 5. CONMIN sends the new and receives the new OBJ from the black box. 6. CONMIN goes to step 3. Steps 3 through 6 are repeated as man times as needed, as specified by the user. 7. The monitor program prints out the final airfoil shape corresponding to the final airfoil design variable values, and stops. The design cp and the actual cp are sometimes printed out to ensure that the final airfoil does generate cp values close to the design values.
x=0
P y=0
P1
What combination of x and y will produce the lowest F(x,y) Subject to the constraints x > 0 , y > 0, x + y < 1 ?
Strategy
• Start at a point P. Compute F(x,y) at P • Take a small step to the right, find F(x+∆x, y) • Use this to find ∂F/ ∂x at P. • Similarly find ∂F/ ∂y at P • Construct a gradient vector . • Find new values of x,y
Alternate Design Variables
y(x) = ybase airfoil (x) + ∑ a n Pn (x)
n=1
TERM P(1) EQUATION
a(12) 40x
N
10.(1.−x)x
P(2)
/e
EXPONE NT j
TABLE n 0.5757166 0.7564708 1.0 1.356915 1.943358 3.106283 6.578813
10.(1.−x)x
P(3)
a(12)
/e
20x
4 5 6 7 8 9
x (1− x) / e
P(J), J=4,10
3x
sin5 xn j
x10
( )
P(11)
10
(Reference : NACA CR 3065 “Analysis of a Theoretically Optimized Transonic Airfoil”)
Airfoil coordinates as Design Variables
• In an optimization problem, the design variables are the quantities whose values are adjusted until the objective function OBJ is minimized. • In an airfoil design by optimization techniques, the obvious choice for design variables are the airfoil y-ordinates at certain fixed number of xlocations. • One can adjust these individual y-ordinates until the objective function OBJ is minimized. • Problem is… there are usually 100 or more design variables if we define the airfoil with 100 nodes. This is too many!
Input : Design Variables OBJ & Constraint Info CONMIN Subroutines User-written Monitor Program OBJ & Constraint Info Input : Design Variables
Black Box
Commonly Used Optimization Tools
• MATLAB has built-in functions! • Vanderplaats at NASA Ames Research Center, developed a computer code is called CONMIN (Constrained Minimization); Reference NASA TMX62282, August 1973). • NASA’s QNMDIF code which uses a similar methodology coupled with a QuasiNewton iterative scheme.
Airfoil Design
Available Approaches
• Optimization Methods • Inverse Design Methods • Ad Hoc “Cut and Try Methods”
Optimization Theory I
f(x,y) P2 x+y=1 x y Black Box
∂F F ( x + ∆x, y ) − F ( x) = ∆x ∂x ∂F F ( x, y + ∆y ) − F ( x) = ∂y ∆y r ∂F r ∂F r ∇F = e1 + e2 ∂x ∂y ∂F ∂ x new = xP − r x δ ∇F ∂F ∂y = yP − r δ ∇F
y new
Black Box
Input a(1) a(2) . . . . a(12) M,α Preprocessor New Airfoil Shape A n a l y s i s c p, cl, cd
Computes New Airfoil Shape
Post Processor
M,α
OBJ
OVERALL FLOW OF INFORMATION