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P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical.

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Presentation on theme: "P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical."— Presentation transcript:

1 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Bezier Functions From Airfoils to the Inverse Problem

2 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering One Dimensional Example Closing DJIA between Aug and Dec 2007 A Bezier function over all the data Order of function = 20 Mean original data = Mean Bezier data = Avg. Error = Maximum Data = Std. Dev (original) = Std. Dev. (Bezier) =

3 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering What is a Bezier Function ? 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

4 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Matrix Description of Bezier Function This allows the use of Array Processing for shorter computer time 3

5 P. Venkataraman Rochester Institute of TechnologyGraduate 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 y a,i, or [Y], find the coefficient matrix, [B] so that the corresponding data set y b,i, [Y B ] produces the least sum of the squared error 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

6 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Bezier Airfoils 6 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.

7 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Airfoil Optimization 7 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 C L Minimum C D Maximum C L /C D Maximum C L 3/2 /C D

8 P. Venkataraman Rochester Institute of TechnologyGraduate 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. V r, V θ, V z, are the velocity components. p is the pressure, ν, the dynamic viscosity. The continuity and the Navier-Stokes equations are Navier-Stokes equation Boundary Conditions : Flow Over a Rotating Disk

9 P. Venkataraman Rochester Institute of TechnologyGraduate 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

10 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering 10Differential Equations - ODEFlow 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 :

11 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering 11Differential Equations - ODEFlow Over a Rotating Disk Bezier Solution : Comparison of Bezier Solution with Numerical Solution

12 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Differential Equations - PDEFlow 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 U 0. 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, ρ is the fluid density and ν is the fluid kinematic viscosity. L 1 is the length of the channel. L 2 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

13 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Differential Equations - PDEFlow 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: 13

14 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Differential Equations - PDEFlow in a Channel Bezier Solution : The solution presented corresponds to m = 9 and n = 6 u velocityv velocity 14

15 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Differential Equations - PDEFlow in a Channel Bezier Solution : The solution presented corresponds to m = 9 and n = 6 p - solution All of the solutions can be represented by explicit polynomials in two parameters– which has not be done before 15

16 P. Venkataraman Rochester Institute of TechnologyGraduate 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 16

17 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Matrix Form of Bezier Function in 3D 18

18 P. Venkataraman Rochester Institute of TechnologyGraduate 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, [B U ] so that the corresponding approximate data [U B ] generates the least value for the sum of the squared error over the data array 18

19 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Three Dimensional Bezier Function – Smooth Data Original Data about 2600 points based on MATLAB Peaks function 3D View of the Data Using the Bezier Filter Contour Plot 3D Plot originalBezier mean std. dev maximum minimum average error: 6.91e-02 19

20 P. Venkataraman Rochester Institute of TechnologyGraduate 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 Less dominant peaks diffused 3D plot Bezier Filter Contour plot 3D plot average error: 6.54e-01 originalBezier mean std. dev maximum minimum

21 P. Venkataraman Rochester Institute of TechnologyGraduate 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 True image processing in MATLAB Bezier filter applied to Red, Green and Blue color separately and combined Highly nonlinear color distribution 21

22 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Single Bezier Functions for the Image Size = 671 KB Bezier function representation Function order 20 x 20 Coefficient storage = 11 KB (3 color streams) Original image 22

23 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Bezier Function in Four Quadrants Original Image 671 KB Four quads Bezier function representation Function order 20 x 20 Coefficient storage = 4*11 KB (3 color streams) = 44 KB 23

24 P. Venkataraman Rochester Institute of TechnologyGraduate 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 [ x i, y i ], i = 1,2,.. p is known as the solution to Then find f(D) and y 0 OR f(D) may be a linear or a nonlinear operator The 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

25 P. Venkataraman Rochester Institute of TechnologyGraduate 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

26 P. Venkataraman Rochester Institute of TechnologyGraduate 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

27 P. Venkataraman Rochester Institute of TechnologyGraduate 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 y a,i, or [Y], find the coefficient matrix, [B] so that the corresponding data set y b,i, [Y B ] produces the least sum of the squared error 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

28 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Step 2:The Generic Form of ODE Many 3 rd order ODE generic forms are used in the paper. For example 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 Linear Generic Form Nonlinear Generic Form 28

29 P. Venkataraman Rochester Institute of TechnologyGraduate 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 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

30 P. Venkataraman Rochester Institute of TechnologyGraduate 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 Enumeration all of the values for the unknowns are considered in combination before the optimum is determined 2.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 1.0*10 5 cpu seconds on a Linux Opteron running MATLAB 2007a 30

31 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Example 1 (Step 1) Best order of Fit (based on y-data) : 14 Number of data points: 200 Sum of Absolute Error (y): e-005 Sum of Squared Error (y): e-011 Average Error (y): e-007 Sum of Absolute Error (x): e-007 Sum of Squared Error (x): e-015 Average Error (x): e-009 Typeoriginal dataBezier data x (initial)11 x (final)55 y (initial)11 y (final)22 dy/dx (initial)not given dy/dx (final)not given d 2 y/dx 2 (initial) not given The original data is discrete x-y data The derivatives are those predicted for the data 31

32 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Example 1 (Step 2) The exponents and coefficients are drawn from the set of three except for h 1 that will belong to a set of 9 values Solution : Exhaustive Enumeration a 1 = 0, a 2 = 1, a 3 = 1, a 4 = 1, b 1 = 1, b 2 = 0, b 3 = 0, c 1 = 1, c 2 = 0, d 1 = 1. The solution for the exponents: The solution for the coefficients: e 1 = 1, e 2 = 1, e 3 = 1, f 1 = 0, f 2 = 0, f 3 = 1, g 1 = 0, g 2 = 1, g 3 = 0, h 1 = h 2 = 0, h 3 = 1. The differential equation This was the same differential equation used to generate the discrete data 32

33 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Example 2 (Step 1) Best order of Fit (based on y-data): 12 No. of data points: 101 Sum of Absolute Error (y): e-005 Sum of Squared Error (y): e-010 Average Error (y): e-007 Sum of Absolute Error (x): e-008 Sum of Squared Error (x): e-018 Average Error (x): e-010 Typeoriginal dataBezier data x (initial) e-013 x (final) y (initial) e-007 dy/dx (initial) e-005 dy/dx (final) d 2 y/dx 2 (initial) The discrete data is created by numerical integration using derivative information The Bezier data approximates the derivative nicely 33

34 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Example 2 (Step 2) Solution : Exhaustive Enumeration The solution for the exponents: The solution for the coefficients: The differential equation A constant nonlinear generic form is used (to reduce time of computation) a 1 = 1, a 2 = 1, a 3 = 2, a 4 = 0, b 1 = 2, b 2 = 2, b 3 = 1, c 1 = 0, c 2 = 0, d 1 = 0 e 1 = 1, e 2 = 0.5, e 3 = 0.5, e 4 = 0.5 This is the Blasius equation used to generate data 34

35 P. Venkataraman Rochester Institute of TechnologyGraduate 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 to coupled ODE single and coupled PDE non smooth data are planned for the future 35

36 P. Venkataraman Rochester Institute of TechnologyGraduate 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

37 P. Venkataraman Rochester Institute of TechnologyGraduate Seminar, January, 7, 2010 Bezier Functions : From Airfoils to the Inverse Problem Mechanical Engineering Questions ??


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