Bezier Functions From Airfoils to the Inverse Problem

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Bezier Functions From Airfoils to the Inverse
Problem
P. Venkataraman
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One Dimensional Example
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Closing DJIA between Aug and Dec 2007
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
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What is a Bezier Function ?
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A Bezier function is a Bezier curve that behaves
like a function
The Bezier curve is defined using a parameter
Instead of yf(x) both x and y depend on the
same parameter value x x(p) and y y(p)
p parameter
Bernstein basis
Number of vertices 5 Order of the function 4
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Matrix Description of Bezier Function
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This allows the use of Array Processing for
shorter computer time
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The Best Bezier Function to fit the Data
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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
Minimize
FOC
Once the coefficient matrix is known, all other
information can be generated using array
processing
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Bezier Airfoils
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There are 2 curves for the top surface 
There are 2 curves for the bottom surface 
All curves are 6 th order 
Properties
Slope continuity is enforced at all curve
junctions (except off course the leading and
trailing edge) 
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.
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Airfoil Optimization
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Single and Multipoint Airfoil Design
Single-Point design cruise Multi-Point
(Two-Point) design
cruise and takeoff.
The airfoil geometry is parameterized using
Bezier Curves The 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

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Differential Equations - ODE
Flow Over a Rotating Disk
A disk of radius R is rotating with the angular
speed ? 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, ?, z) is used for description. Vr, V?, Vz,
are the velocity components.  p is the pressure,
?, the dynamic viscosity.  The continuity and the
Navier-Stokes equations are
Navier-Stokes equation
Boundary Conditions
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Differential Equations - ODE
Flow Over a Rotating Disk
Transformation Relations
Transformed Navier-Stokes equation
Boundary Conditions
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Differential 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
Minimize
Subject to
Solution
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Differential Equations - ODE
Flow Over a Rotating Disk
Bezier Solution
Comparison of Bezier Solution with Numerical
Solution
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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
Navier-Stokes Equations
Boundary Conditions
In the above, ? is the fluid density and ? 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.
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Differential Equations - PDE
Flow in a Channel
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
Boundary Conditions
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Differential Equations - PDE
Flow in a Channel
Bezier Solution
The solution presented corresponds to m 9 and n
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u velocity
v velocity
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Differential Equations - PDE
Flow in a Channel
Bezier Solution
The solution presented corresponds to m 9 and n
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p - solution
All of the solutions can be represented by
explicit polynomials in two parameters which has
not be done before
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Bezier Function in 3D
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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
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Matrix Form of Bezier Function in 3D
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Bezier Filter for 3D Data
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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
Minimize
FOC
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
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Three Dimensional Bezier Function Smooth Data
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Original Data about 2600 points based on MATLAB
Peaks function
3D View of the Data
Using the Bezier Filter
Contour Plot
3D Plot
average error 6.91e-02
original Bezier
mean 0.317 0.312
std. dev. 1.116 1.086
maximum 8.042 7.360
minimum -6.521 -6.405
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Three Dimensional Bezier Function Rough Data
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Same peaks function but randomly perturbed on
both sides
Less dominant peaks diffused
3D plot
Bezier Filter
Contour plot
3D plot
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
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Bezier Function in Image Handling
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The original image is 960 x 1280 pixels of size
671 KB
True image processing in MATLAB
Bezier filter applied to Red, Green and Blue
color separately and combined
Highly nonlinear color distribution
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Single Bezier Functions for the Image
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Original image
Bezier function representation
Size 671 KB
Function order 20 x 20
Coefficient storage 11 KB (3 color streams)
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Bezier Function in Four Quadrants
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Original Image 671 KB
Bezier function representation
Four quads
Function order 20 x 20
Coefficient storage 411 KB (3 color streams)
44 KB
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The Inverse ODE Problem
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The inverse problem in this paper is very direct
find the differential equation and the
boundary conditions if the discrete solution is
known everywhere
OR
If xi, yi, i 1,2, .. p is known as the
solution to
Then find f(D) and y0
f(D) may be a linear or a nonlinear operator The
ODE is homogenous
after all
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
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The Solution Process
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• 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

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Why a Bezier Function?
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Bezier functions are parametric curves based on
Bernstein polynomial basis functions
Bezier functions can provide explicit solutions
to the forward boundary value problem very
effectively The authors 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
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
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Step 1The Best Bezier Function to fit the Data
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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
Minimize
FOC
Once the coefficient matrix is known, all other
information, including the derivatives can be
generated using array processing
The best m is determined by the lowest value of E
This is Step 1 of the solution process
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Step 2The Generic Form of ODE
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Many 3rd order ODE generic forms are used in the
paper. For example
Linear Generic Form
There are two types of unknowns the exponents
of the derivatives the coefficients multiplying
the terms The exponents are expected to be
integers The coefficients are unrestricted The
function and its derivatives are known quantities
after Step 1
Nonlinear Generic Form
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Establishing the Unknowns
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A Least Squared Error Technique is used to
determine the unknowns
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
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Discrete Programming Used in The Paper
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Two procedures are considered in this paper
1. Exhaustive Enumeration all of the values
for the unknowns are considered in combination
before the optimum is determined

• Simple Heuristic Programming
• simple 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
1.0105 cpu seconds on a Linux Opteron running
MATLAB 2007a
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Example 1 (Step 1)
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Best order of Fit (based on y-data) 14 Number
of data points 200
Sum of Absolute Error (y) 7.27217e-005 Sum of
Squared Error (y) 3.96250e-011 Average Error
(y) 3.63608e-007 Sum of Absolute
Error (x) 4.56362e-007 Sum of Squared Error
(x) 2.05982e-015 Average Error (x)
2.28181e-009
Type original data Bezier data
x (initial) 1 1
x (final) 5 5
y (initial) 1 1
y (final) 2 2
dy/dx (initial) not given -7.2728
dy/dx (final) not given 2.1184
d2y/dx2 (initial) not given 6.5163
The original data is discrete x-y data The
derivatives are those predicted for the data
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Example 1 (Step 2)
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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
The solution for the exponents
a1 0, a2 1, a3 1, a4 1, b1 1, b2 0,
b3 0, c1 1, c2 0, d1 1.
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. 
The differential equation
This was the same differential equation used to
generate the discrete data
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Example 2 (Step 1)
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Best 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 data
x (initial) 0 1.3234e-013
x (final) 6 6.0000
y (initial) 0 1.9390e-007
dy/dx (initial) 0 -1.3919e-005
dy/dx (final) 1 1.0001
d2y/dx2 (initial) 0.3326 0.3329
The discrete data is created by numerical
integration using derivative information
The Bezier data approximates the derivative nicely
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Example 2 (Step 2)
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A constant nonlinear generic form is used (to
reduce time of computation)
Solution Exhaustive Enumeration
The solution for the exponents
a1 1, a2 1, a3 2, a4 0, b1 2, b2 2,
b3 1, c1 0, c2 0, d1 0
The solution for the coefficients
e1 1, e2 0.5, e3 0.5, e4 0.5
The differential equation
This is the Blasius equation used to generate data
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Work in Process
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The computation time is a serious issue for a
broader range of values. Global optimization
techniques may provide a relief
Extension to coupled ODE single and coupled
PDE non smooth data are planned for the future
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Properties of the Bezier Function
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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
Gradient and derivative information of the data
are easy to obtain
Bezier functions naturally decouples the
independent and the dependent variables
Bezier functions coupled with optimization can
solve all kinds of mathematical problems
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Questions ??