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Transcript of P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions...
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Bezier Functions From Airfoils to the Inverse Problem
P. Venkataraman
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
One Dimensional Example
0 50 100 150 200 2501.2
1.25
1.3
1.35
1.4
1.45x 10
4
Ori
gin
al D
ata,
Fit
ted
Dat
a
points
Closing DJIA between Aug and Dec 2007
0 50 100 150 200 2501.2
1.25
1.3
1.35
1.4
1.45x 10
4
Ori
gin
al D
ata,
Fit
ted
Dat
a
points
DJIA - Adjust Close 17 Sep - Dec 18
A Bezier function over all the data
Order of function = 20
Mean original data = 13172.432
Mean Bezier data = 13172.423
Avg. Error = 98.34
Maximum Data = 14164.53
Std. Dev (original) = 530.19
Std. Dev. (Bezier) = 514.68
1
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
What is a Bezier Function ?
0 1 2 3 4 50
0.5
1
1.5
2
2.5
3
3.5
4
x: independent variable
y: d
ep
end
en
t va
ria
ble
s
[a1,b1][a1,b1]
[a2,b2]
[a3,b3]
[a4,b4]
[a5,b5]
Convex hullBezier VerticesBezier Curve: order 4
,0
( ) ( ) ( ) , 0 1
n
i n ii
Bx p y p J p p
1, ( ) ( )i n in i
nJ p p p
i
p : parameter
Bernstein basis
Number of vertices: 5
Order of the function : 4
A Bezier function is a Bezier curve that behaves like a function
The Bezier curve is defined using a parameter
Instead of y=f(x);
both x and y depend on the same parameter value; x = x(p) and y = y(p)
2
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Matrix Description of Bezier Function
0 1 2 3 4 50
0.5
1
1.5
2
2.5
3
3.5
4
x: independent variable
y: d
ep
end
en
t va
ria
ble
s
[a1,b1][a1,b1]
[a2,b2]
[a3,b3]
[a4,b4]
[a5,b5]
Convex hullBezier VerticesBezier Curve: order 4
[ ( ) ( )] [ ][ ][ ]x p y p P N B
4 3 2[ ] [ 1];
1 -4 6 -4 1 0 0
-4 12 -12 4 0 1 3
[ ] 6 -12 6 0 0 [ ] 2 1
-4 4 0 0 0 3 2
1 0 0 0 0 5
P p p p p
N B
0
This allows the use of Array Processing for shorter computer time
3
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
For a selected order of the Bezier function (n) Given a set of (m) vector data ya,i , or [Y], find the coefficient matrix, [B] so that the corresponding data set yb,i , [YB ] produces the least sum of the squared error
2
, ,
m
a i b ii
E y y T T
B B A AE Y Y Y Y Y P NB Y P NB
0E
B
1[ ] [ ] [ ]T TA A AB P P P Y
Minimize
FOC:
The Best Bezier Function to fit the Data
Once the coefficient matrix is known, all other information can be generated using array processing
4
0 50 100 150 200 2501.2
1.25
1.3
1.35
1.4
1.45x 10
4
Ori
gin
al D
ata,
Fit
ted
Dat
a
points
DJIA - Adjust Close 17 Sep - Dec 18
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Bezier Airfoils 6
x/c
y/c
01
curve 1 curve 2
curve 3 curve 4
There are 2 curves for the top surface
There are 2 curves for the bottom surface
All curves are 6 th order
Slope continuity is enforced at all curve junctions (except off course the leading and trailing edge)
Properties
Second derivative continuity is enforced between the forward and rear curves
Second derivative direction continuity is enforced at the leading edge
Any past/contemporary/ single element airfoil, low speed or transonic, can be constructed using the Bezier Curves shown above.
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Airfoil Optimization 7
x/c
y/c
01
curve 1 curve 2
curve 3 curve 4
Single and Multipoint Airfoil Design
Single-Point design: cruise Multi-Point (Two-Point) design –
cruise and takeoff.
The airfoil geometry is parameterized using Bezier CurvesThe aerodynamic information is obtained using the XFOIL program(Professor Drela MIT)
Airfoils can be designed for geometry• Area • Maximum thickness • Maximum thickness for top and bottom• Location of maximum thickness • Disparate locations of maximum thickness
Airfoils can be designed for performance• Maximum CL
• Minimum CD
• Maximum CL/CD
• Maximum CL3/2 /CD
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
8Differential Equations - ODE
A disk of radius R is rotating with the angular speed ω in still fluid. The flow is steady, incompressible, has constant property, and is axisymmetric. The fluid at the disk has to satisfy the no slip condition. The centrifugal effects cause the fluid to leave the disk radially near the disk. The flow above the disk must replace this airflow through a downward spiraling flow. A cylindrical coordinate system (r, θ, z) is used for description. Vr, Vθ, Vz, are the velocity components. p is the pressure, ν, the dynamic viscosity. The continuity and the Navier-Stokes equations are
2 2 2
2 2
2 2
2 2
2 2
2 2
0
1
1 1
zr r
zr r r rr z
rr z
z z z z zr z
VV Vr r z
VVV p V V VV V
r r z r r r r z
V V V V V V VV V
r r z r r r z
V V V V VpV V
r z z r r r z
0: 0; ; 0;
: 0; 0;r z
r
z V V r V
z V V
Navier-Stokes equation
Boundary Conditions :
Flow Over a Rotating Disk
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
9Differential Equations - ODE
Transformation Relations
Boundary Conditions :
Flow Over a Rotating Disk
Transformed Navier-Stokes equation
; ( ); ( );
( ); ( ) ( );
r
z
zZ z V r F Z V r G Z
V H Z p z P Z
2 2
2 0
0
2 0
0
F H
F FH G F
FG HG G
P HH H
0; 0; 1; 0; 0
( 6); 0; 0
Z F G H P
Z F G
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
10Differential Equations - ODE Flow Over a Rotating Disk
Bezier Solution :
Three Bezier functions will be used to identify the functions F, G, and H. This is now a coupled set of nonlinear differential equations.
Optimization Problem :
22 2 2
1
1002
1
2
2
pn
i i i i i i ii
i i i i ii
f F H F F H G F
FG H G G
Minimize :
Subject to : 1 1 1 1 1 1
1 1 1 1
, [0,0]; , [0,1]; [ , ] [0,0]
, [0,0]; , [0,0];m m m m
Z F Z G Z H
Z F Z G
6 5 4 3 2
6 5 4 3 2
6 5 4 3
( ) -0.36800 - 0.23400 1.1100 0.68000 1.4400 3.3720
( ) 1.5270 - 4.4280 2.7150 3.4200 - 4.9500 1.7160
( ) - 0.56300 3.1740 - 6.7500 6.1000 - 0.8
Z p p p p p p p
F p p p p p p p
G p p p p p
2
6 5 4 3 2
0 1
8500 - 2.0760 1
( ) 1.7270 - 3.8640 - 0.69000 7.7600 - 5.7750
p
p p
H p p p p p p
Solution :
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
11Differential Equations - ODE Flow Over a Rotating Disk
Bezier Solution :
0 1 2 3 4 5 6-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Z
(F,
G,
H)
U - velocity
Curve:U - velocity
V - velocityCurve:V - velocity
W - Velocity
Curve:W - Velocity
-1 -0.5 0 0.5 10
1
2
3
4
5
6Laminar Flow near a Spinning Disk
Z
F, G, H
F (Vr)
F-Bezier
G (V)
G - Bezier
H (Vz)
H - Bezier
Comparison of Bezier Solution with Numerical Solution
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Differential Equations - PDE Flow in a Channel
A steady, two-dimensional, constant property flow takes place in a two dimensional channel. The x-velocity (u) at the inlet is constant with the value U0. There is no y-velocity (v) at the inlet. The no slip conditions apply on both wall
x
y
L1
L2
U0 u = 0, v =0
u = 0, v =0
v = 0Navier-Stokes Equations :
2 2
2 2
2 2
2 2
0
1
1
continuity:
x-momentum:
y-momentum:
u v
x y
u u p u uu v
x y x x y
v v p v vu v
x y y x y
Boundary Conditions :
0
2 2 2
0 0 0 0 0
0 0 0 0 0
0 0
; ( , ) ( , ) ; ( , ) ( , ) ; ( , ) ( , ) ;
; ( , ) ( , ) ; ( , ) ( , ) ;
; ( , ) ( , ) ; ( , ) ( , ) ;
x u x y u y U v x y v y p x y p y c
y u x y u x v x y v x
y L u x y u x L v x y v x L
In the above, ρ is the fluid density and ν is the fluid kinematic viscosity. L1 is the length of the channel. L2 is the width of the channel. The domain is called the entering region of the flow as the viscous effects through the walls will shape the velocity profile in the channel as the flow proceeds left to right.
12
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Differential Equations - PDE Flow in a Channel
x
y
L1
L2
U0 u = 0, v =0
u = 0, v =0
v = 0
The nonlinear BVP problem will be solved using Bezier functions.
Here the solution will be represented by three surfaces in the solution domain.
The first is the solution for the velocity in the x-direction u(x, y), the second is the solution for the velocity in the y-direction v(x, y), and the third one is the solution for the pressure p(x ,y).
The Optimization Problem :
2
1 1
22 2
2 21 1
22 2
2 21 1
1
1
, ,
, ,, ,
, , , ,
( )p q
i j i j i j
p q
i j i j i ji j i j
p q
i j i j i j i j i j
u vMinimize F X
x y
u u p u uu v
x y x x y
v v p v vu v
x y y x y
0
2 2 2
0 0 0 0
0 0 0 0 0 0 0
0 0 0 0
; ( , ) ; ( , ) ;
; ( , ) ; ( , ) ;
; ( , ) ; ( , ) ;
x u y U v y
y u x v x
y L u x L v x L
Boundary Conditions:
13
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Differential Equations - PDE Flow in a Channel
x
y
L1
L2
U0 u = 0, v =0
u = 0, v =0
v = 0
Bezier Solution :
The solution presented corresponds to m = 9 and n = 6
0.33
0.661
0.991
1.32
1.65
1.98
2.31
2.64
2.97
3.3
3.63
3.96
4.294.62
4.95
5.28
5.61
5.95
6.28
6.61
4.62
4.95
y
x
Contour for u
0 10 20 30 400
0.2
0.4
0.6
0.8
1-0.0416-0.0328-0.024-0.0152
-0.00636
-0.00197-0.00197
0.00243
-0.00636
0.00683
0.0112
-0.0108
0.0156
-0.0152-0.0196
0.02
-0.024
0.0244
-0.0284
0.0288
-0.0328
0.0332
-0.0372
0.0376
-0.00197
0.00243
-0.0416
0.042 -0.00197
y
x
Contour for v
0 10 20 30 400
0.2
0.4
0.6
0.8
1
0 50
0.2
0.4
0.6
0.8
1
y
u
Profile for u
x =0
x =2.25
x =4.5
x =6.75
x =11.25
x =18
x =22.5
x =31.5
x =36
x =45-0.05 0 0.050
0.2
0.4
0.6
0.8
1
y
v
Profile for v
x =0
x =2.25
x =4.5
x =6.75
x =11.25
x =18
x =22.5
x =31.5
x =36
x =45
u velocity v velocity
14
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Differential Equations - PDE Flow in a Channel
x
y
L1
L2
U0 u = 0, v =0
u = 0, v =0
v = 0
Bezier Solution :
The solution presented corresponds to m = 9 and n = 6
p - solution
-1.41
1.3
4
6.71
9.42
12.1
14.8
17.5
20.2
22.9
All of the solutions can be represented by explicit polynomials in two parameters– which has not be done before
15
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Bezier Function in 3D
A 3D Bezier function will be a surface in 2D. Bezier surface can be described as a vector-valued function of two parameters r and s
[ ( , ) ( , ) ( , )]; 0 , 1x r s y r s u r s r s
, ,
, , ,0 0
( ) (1 ) ; 0 1; ( ) (1 ) ; 0 1
[ ( , ) ( , ) ( , )] ( , ) ( ) ( )
i m i j n j
m i n j
m n
i j m i n ji j
m nJ r r r r K s s s s
i j
x r s y r s u r s Q r s B J r K s
2 2
4 3 2
3 4 4
3 2 3
3 2 2 3 2
( , ) -1 3 3
( , ) - 4 8 - 6 8
( , ) (-1 48 - 78 12 )
(4 -112 180 - 24 )
(-6 60 - 90 ) (-8 12 8)
0 1 0 1
x r s r r r
y r s s s s s
u r s r r r s
r r r s
r r s r r s
r s
16
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Matrix Form of Bezier Function in 3D
[ ( , ) ( , ) ( , )]; 0 , 1x r s y r s u r s r s
[ ( , ) ( , ) ( , )] ( , )T
x r s y r s u r s Q r s R M B N S
4 3 21[ ] [ ... ] 1TTn nS s s s s s s s
1 3 2 1[ ...1] 1m mR r r r r r
1 0
1 2
1 0
0
11 1 ... 1
0 0 1 1
11 1 ... 0
0 1 1 2
[ ] . . . 0
11 1 ... 0
0 1 1 0
1 0 ... 00 0
m m
m m
m m m m m m m
m m m m
m m m m
m m
M
m m m m
m m
[ ]M
-1 3 -3 1
3 -6 3 0
-3 3 0 0
1 0 0 0
[ ]N
1 -4 6 -4 1
-4 12 -12 4 0
6 -12 6 0 0
-4 4 0 0 0
1 0 0 0 0
[ ]B
[0 0 0] [0 2 2] [0 3 3] [0 5 4] [0 6 5]
[1 0 0] [1 2 2] [1 3 3] [1 5 2] [1 6 1]
[3 0 0] [3 2 3] [3 3 0] [3 5 3] [3 6 5]
[5 0 0] [5 2 3] [5 3 0] [5 5 3] [5 6 5]
18
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Minimize
FOC:
Bezier Filter for 3D Data
Once the coefficient matrix is known, all other information can be generated using array processing
For the filter, the best order is chosen on minimum absolute error
Given a set of array data [U], assuming an order for each dimension (m, n),
find the Bezier function coefficient matrix, [BU] so that the corresponding approximate data [UB] generates the least value for the sum of the squared error over the data array
2B
i j
E U U TTB A U A A U AU R M B N S F B G
0U
E
B
1T TU A A A A AB G IF F G IF U[ ] [ ] [ ]
18
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Three Dimensional Bezier Function – Smooth Datay
x
Original Data
5 10 15 20 25 30 35 40 45 50
5
10
15
20
25
30
35
40
45
50
-6
-4
-2
0
2
4
6
Original Data about 2600 points based on MATLAB Peaks function
3D View of the Data
010
2030
4050
60
0
10
20
30
40
50
60-8
-6
-4
-2
0
2
4
6
8
10
x
Original Data
y
Ori
gin
al
-6
-4
-2
0
2
4
6
8
y
x
m =12, n =15 ,Least Sum of Absolute Error :179.8217
5 10 15 20 25 30 35 40 45 50
5
10
15
20
25
30
35
40
45
50
-6
-4
-2
0
2
4
6
Using the Bezier Filter
010
2030
4050
60
0
10
20
30
40
50
60-8
-6
-4
-2
0
2
4
6
8
x
Bezier Data
y
Bez
ier
-6
-4
-2
0
2
4
6 Contour Plot
3D Plot
original Bezier
mean 0.317 0.312
std. dev. 1.116 1.086
maximum 8.042 7.360
minimum -6.521 -6.405
average error: 6.91e-02
19
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Three Dimensional Bezier Function – Rough Data
Same peaks function but randomly perturbed on both sides
y
x
Original Data
5 10 15 20 25 30 35 40 45 50
5
10
15
20
25
30
35
40
45
50
-6
-4
-2
0
2
4
6
Less dominant peaks diffused3D plot
010
2030
4050
60
0
10
20
30
40
50
60-8
-6
-4
-2
0
2
4
6
8
10
x
Original Data
y
Ori
gin
al
-6
-4
-2
0
2
4
6
8 Bezier FilterContour plot
y
x
m =12, n =12 ,Least Sum of Absolute Error :1702.726
5 10 15 20 25 30 35 40 45 50
5
10
15
20
25
30
35
40
45
50
-6
-4
-2
0
2
4
6
3D plot
010
2030
4050
60
0
10
20
30
40
50
60-8
-6
-4
-2
0
2
4
6
8
x
Bezier Data
y
Bez
ier
-6
-4
-2
0
2
4
6
average error: 6.54e-01
original Bezier
mean 0.322 0.325
std. dev. 0.859 1.035
maximum 8.253 7.481
minimum -7.651 -6.565
20
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Bezier Function in Image Handling
The original image is 960 x 1280 pixels of size 671 KB
200 400 600 800 1000 1200
100
200
300
400
500
600
700
800
900
True image processing in MATLAB
Bezier filter applied to Red, Green and Blue color separately and combined
Highly nonlinear color distribution
21
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Single Bezier Functions for the Image
200 400 600 800 1000 1200
100
200
300
400
500
600
700
800
900
200 400 600 800 1000 1200
100
200
300
400
500
600
700
800
900
Size = 671 KB
Bezier function representation
Function order 20 x 20
Coefficient storage = 11 KB (3 color streams)
Original image
22
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Bezier Function in Four Quadrants
Original Image 671 KB
200 400 600 800 1000 1200
100
200
300
400
500
600
700
800
900
Four quads
Bezier function representation
200 400 600 800 1000 1200
100
200
300
400
500
600
700
800
900
Function order 20 x 20
Coefficient storage = 4*11 KB (3 color streams) = 44 KB
23
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
The Inverse ODE Problem
The inverse problem in this paper is very direct :
find the differential equation and the boundary conditions if the discrete solution is known everywhere
If [xi, yi], i = 1,2, .. p is known as the solution to
0( ) 0; (0)f D y y y
Then find f(D) and y0
OR
f(D) may be a linear or a nonlinear operatorThe ODE is homogenous
the forward or the direct boundary value problem is the determination of the solution everywhere if the differential equation is known and the boundary conditions are given
after all
24
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
The Solution Process
The procedure involves two steps:
Step 1: A “best” Bezier function is fitted to the data
This function, which is also the solution to the ODE, will satisfy the differential equation and identify the boundary condition
Step 2: The specific form of the differential equation is determined
This form is established from a generic representation of the ODE using a set of exponent and coefficient values
25
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Why a Bezier Function?
Bezier functions can provide explicit solutions to the forward boundary value problem very effectively
The author’s papers in previous CIE conferences have shown Bezier functions can solve linear or nonlinear, single or multi variable, ordinary or partial differential equations, with initial and/or boundary values
Bezier functions are parametric curves based on Bernstein polynomial basis functions
“the Bernstein polynomial approximation to a continuous function mimics the gross features of the function remarkably well” - Gordon and Riesenfeld
As the order of the polynomial is increased, this approximation converges uniformly to the function and its derivatives where they exist
The Bezier curve delivers, at the minimum, the same smoothness as the primitive function it is trying to emulate
26
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
For a selected order of the Bezier function (n) Given a set of (m) vector data ya,i , or [Y], find the coefficient matrix, [B] so that the corresponding data set yb,i , [YB ] produces the least sum of the squared error
2
, ,
m
a i b ii
E y y
T T
B B A AE Y Y Y Y Y P NB Y P NB
0E
B
1[ ] [ ] [ ]T TA A AB P P P Y
Minimize
FOC:
Step 1:The Best Bezier Function to fit the Data
Once the coefficient matrix is known, all other information, including the derivatives can be generated using array processing
This is Step 1 of the solution process
The best m is determined by the lowest value of E
27
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Step 2:The Generic Form of ODE
Many 3rd order ODE generic forms are used in the paper. For example
There are two types of unknowns: the exponents of the derivativesthe coefficients multiplying the terms
The exponents are expected to be integers The coefficients are unrestrictedThe function and its derivatives are known quantities after Step 1
Linear Generic Form
1 2 3
1 2
11
1
3 22 4
1 2 3 3 2
232
1 2 3 2
21 2 3
21 2 3
( )
( )
( )
( ) 0
a a aa
b bb
cc
d
d y d y dye e x e x y
dxdx dx
d y dyf f x f x y
dxdx
dyg g x g x y
dx
h h x h x y
Nonlinear Generic Form
1 1
11
3 22 2
1 2 3 1 2 33 2
2 21 2 3 1 2 3
( ) ( )
( ) ( ) 0
a b
cd
d y d ye e x e x f f x f x
dx dx
dyg g x g x h h x h x y
dx
28
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Establishing the Unknowns
A Least Squared Error Technique is used to determine the unknowns
1 2 13 2 1
4 3 2 1
23 2 2
1 2 3 43 2 21
( )
a a ba b cN a b c di i i i
i i ii i i i
Minimize F
d f d f df d f df dfA f A f A f A f
dx dx dxdx dx dx
N: the number of data points
This is the objective for linear constant coefficient form A similar one can be used for the generic nonlinear form
A continuous application of standard optimization technique was unsuccessful because the exponents were not integers
A mixed integer (exponents) – continuous (coefficients) approach was also unsuccessful because the solution will determine trivial values
Solution was only possible through discrete programming
29
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Discrete Programming Used in The Paper
Two procedures are considered in this paper
1. Exhaustive Enumerationall of the values for the unknowns are considered in combination before the
optimum is determined
2. Simple Heuristic Programmingsimple heuristic exhaustive enumeration over predetermined number of cycles
(1 billion)
Discrete Programming is incredibly time extensive
For the linear constant coefficient form, allowing 3 values for each unknown required 1.0*105 cpu seconds on a Linux Opteron running MATLAB 2007a
30
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Example 1 (Step 1)
1 1.5 2 2.5 3 3.5 4 4.5 5-5
-4
-3
-2
-1
0
1
2
y -
Ori
gin
al D
ata
x0 1 2 3 4 5 6-5
-4
-3
-2
-1
0
1
2
3
y -
Ori
gin
al D
ata,
Fit
ted
Dat
a
x
original data
Bezier approximationBest order of Fit (based on y-data) : 14Number of data points: 200 Sum of Absolute Error (y): 7.27217e-005Sum of Squared Error (y): 3.96250e-011Average Error (y): 3.63608e-007Sum of Absolute Error (x): 4.56362e-007Sum of Squared Error (x): 2.05982e-015Average Error (x): 2.28181e-009
Type original data Bezier datax (initial) 1 1x (final) 5 5y (initial) 1 1y (final) 2 2dy/dx (initial) not given -7.2728dy/dx (final) not given 2.1184d2y/dx2 (initial)
not given 6.5163
The original data is discrete x-y data
The derivatives are those predicted for the data
31
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Example 1 (Step 2)
0 1 2 3 4 5 6-5
-4
-3
-2
-1
0
1
2
3
y -
Ori
gin
al D
ata,
Fit
ted
Dat
a
x
original data
Bezier approximation
1 1
11
3 22 2
1 2 3 1 2 33 2
2 21 2 3 1 2 3
( ) ( )
( ) ( ) 0
a b
cd
d y d ye e x e x f f x f x
dx dx
dyg g x g x h h x h x y
dx
The exponents and coefficients are drawn from the set of three except for h1 that will belong to a set of 9 values
Solution : Exhaustive Enumeration
a1 = 0, a2 = 1, a3 = 1, a4 = 1, b1 = 1, b2 = 0, b3 = 0, c1 = 1, c2 = 0, d1 = 1.The solution for the exponents:
The solution for the coefficients:
e1 = 1, e2 = 1, e3 = 1, f1 = 0, f2 = 0, f3 = 1, g1 = 0, g2 = 1, g3 = 0, h1 = -0.25 h2 = 0, h3 = 1.
22 2
2( 0.25) 0
d y dyx x x y
dxdx
The differential equation
This was the same differential equation used to generate the discrete data
32
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Example 2 (Step 1)
0 1 2 3 4 5 60
0.5
1
1.5
2
2.5
3
3.5
4
4.5
y -
Ori
gin
al D
ata,
Fit
ted
Dat
a
x0 1 2 3 4 5 6 70
0.5
1
1.5
2
2.5
3
3.5
4
4.5
y -
Ori
gin
al D
ata,
Fit
ted
Dat
a
x
Original Data
Bezier ApproximationBest order of Fit (based on y-data): 12 No. of data points: 101 Sum of Absolute Error (y): 8.90327e-005 Sum of Squared Error (y): 1.51515e-010 Average Error (y): 8.81512e-007 Sum of Absolute Error (x): 1.23819e-008 Sum of Squared Error (x): 2.26533e-018 Average Error (x): 1.22593e-010
Type original data Bezier datax (initial) 0 1.3234e-013x (final) 6 6.0000y (initial) 0 1.9390e-007dy/dx (initial) 0 -1.3919e-005dy/dx (final) 1 1.0001d2y/dx2 (initial) 0.3326 0.3329
The discrete data is created by numerical integration using derivative information
The Bezier data approximates the derivative nicely
33
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Example 2 (Step 2)
Solution : Exhaustive EnumerationThe solution for the exponents:
The solution for the coefficients:
The differential equation
0 1 2 3 4 5 6 70
0.5
1
1.5
2
2.5
3
3.5
4
4.5
y -
Ori
gin
al D
ata,
Fit
ted
Dat
a
x
Original Data
Bezier Approximation
1 2 13 24 3
12 1
3 2 2
1 23 2 2
3 4( ) 0
a a ba ba b
cc d
y yd d dy d y dye y e y
dx dxdx dx dx
dye y e y
dx
A constant nonlinear generic form is used (to reduce time of computation)
a1 = 1, a2 = 1, a3 = 2, a4 = 0, b1 = 2, b2 = 2, b3 = 1, c1 = 0, c2 = 0, d1 = 0
e1 = 1, e2 = 0.5, e3 = 0.5, e4 = 0.51 13 2
3 21
d y d y
dy dy
2dy
dx
21 22
20.5
d y dy
dxdy
1
3 2
3 2
0
0.5 0
y
d y d yy
dy dx
This is the Blasius equation used to generate data
34
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Work in Process
The computation time is a serious issue for a broader range of values. Global optimization techniques may provide a relief
Extension tocoupled ODEsingle and coupled PDEnon smooth data
are planned for the future
35
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Bezier filter is easy to incorporate and can work for regular, unpredictable data, and images
The Bezier functions have excellent blending and smoothing properties
High order but well behaved polynomial functions can be useful in capturing the data content and underlying behavior
Bezier functions naturally decouples the independent and the dependent variables
Properties of the Bezier Function
Gradient and derivative information of the data are easy to obtain
36
Bezier functions coupled with optimization can solve all kinds of mathematical problems
P. VenkataramanP. Venkataraman
Rochester Institute of Technology Graduate Seminar, January, 7, 2010
Bezier Functions : From Airfoils to the Inverse Problem
Mechanical Engineering
Questions ??