DETC2008-49101

DETC2008-49101

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DETC2008-49101

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1. CTESA: Computational Technologies for Engineering Sciences Applications ASME Computers and Information in Engineering Conference DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P.Venkataraman Rochester Institute of Technology Department of Mechanical Engineering August 3 - 6, 2008, New York City NY, USA

2. DETC2008-49101 Presentation Outline CTESA: CIE-2-3 Numerical method for modeling 1. Motivation 2. Bezier Function 3. Data Fitting 4. Computational Resource 5. Example 1 Example 2 Example 4 6. Conclusions ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 1/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

3. DETC2008-49101 Motivation Current Status CTESA: CIE-2-3 Numerical method for modeling In prior presentations of this technical committee it has been shown that it is possible to obtain explicit solutions in polynomial form for Linear or nonlinear, Single or coupled, Ordinary or partial, Differential equations using Bezier functions ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 2/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

4. DETC2008-49101 Motivation Inverse Problem CTESA: CIE-2-3 Numerical method for modeling One definition of an inverse problem and its solution is Given a sequence of data Establish the differential equation whose solution Is represented by the data ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 3/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

5. DETC2008-49101 Motivation This Paper CTESA: CIE-2-3 Numerical method for modeling This paper approaches the solution of the inverse problem in two steps Given a sequence of data {xi, yi} Find a function that best fits the data – y(x) Then establish the coefficients of the differential system that the function will belong too: This paper addresses only the first step ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 4/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

6. DETC2008-49101 Bezier Function Description CTESA: CIE-2-3 Numerical method for modeling p : parameter Bernstein basis Number of vertices: 5 Order of the function : 4 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 5/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

7. DETC2008-49101 Bezier Function Matrix Description CTESA: CIE-2-3 Numerical method for modeling Matrix Description ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 6/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

8. DETC2008-49101 Data Fitting Problem Definition CTESA: CIE-2-3 Numerical method for modeling 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: ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 7/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

9. DETC2008-49101 Data Fitting Data Decoupling CTESA: CIE-2-3 Numerical method for modeling The matrix definition for the Bezier function is It can be recognized as And can be decoupled as ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 8/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

10. DETC2008-49101 Computational Resources CTESA: CIE-2-3 Numerical method for modeling Software used: MATLAB 2006b for plots and calculations This work is independent of any language/software/platform 32-bit architecture required limiting the order of the Bezier functions to 20 Standard data statistics is used for comparison ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 9/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

11. DETC2008-49101 Examples CTESA: CIE-2-3 Numerical method for modeling Three of the five examples are presented Example 1: Smooth Data at Equidistant Intervals Example 2: Rough Data at Arbitrary Intervals Example 4: Unorganized Data ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 10/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

12. DETC2008-49101 Examples Example 1 CTESA: CIE-2-3 Numerical method for modeling The data is generated at equidistant intervals of the independent variable (x)The dependent variable (y) values are generated using a smooth function There are 101 data pairs. Best order: 15 Error x: 1.3e-06 Error y: 8.8e-08 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 11/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

13. DETC2008-49101 Examples Example 1- statistics CTESA: CIE-2-3 Numerical method for modeling Comparison of x-data Comparison of y-data ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 12/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

14. DETC2008-49101 Examples Example 1 - Coefficients CTESA: CIE-2-3 Numerical method for modeling Bezier Coefficient Values: The data in Example 1 can be reproduced by a super continuous 15th order Bezier function, whose derivatives can be easily established by known calculations ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 13/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

15. DETC2008-49101 Examples Example 1 – Explicit Polynomial CTESA: CIE-2-3 Numerical method for modeling ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 14/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

16. DETC2008-49101 Examples Example 2 CTESA: CIE-2-3 Numerical method for modeling Random values between specified limits are used to construct values for x and y. It is then sorted in ascending order. Best order: 13 Range x: Error sum : 38.96 Range y: Error sum: 3.31 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman 15/20 Mechanical Engineering Rochester Institute of Technology

17. DETC2008-49101 Examples Example 2- statistics CTESA: CIE-2-3 Numerical method for modeling Data statistics : x Data statistics: y ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 16/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

18. DETC2008-49101 Examples Example 4 CTESA: CIE-2-3 Numerical method for modeling In this example the Adjusted Closing values of the Dow Jones Industrial Average between May 17 and December 18, 2007, is used for the original data Best order: 20 (maximum) Almost all data points are within a standard deviation of the Bezier representation ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 17/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

19. DETC2008-49101 Examples Example 4- statistics CTESA: CIE-2-3 Numerical method for modeling Statistics: y data The Bezier representation preserves the average value of the data ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 18/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

20. DETC2008-49101 Conclusions CTESA: CIE-2-3 Numerical method for modeling Bezier functions are easy to incorporate and can track regular and unpredictable data very well The Bezier functions have excellent blending and smoothing properties High order of functions can be useful in capturing the data content and underlying behavior The mean of the Bezier data is the same as the mean of the original data Bezier functions naturally decouples the independent and the dependent variables ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS 19/20 P. Venkataraman Mechanical Engineering Rochester Institute of Technology

21. DETC2008-49101 Conclusions CTESA: CIE-2-3 Numerical method for modeling Thank You! Questions? ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical Engineering Rochester Institute of Technology