FEM: Introduction and Weighted Residual Methods
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The document covers weighted residual methods for approximating solutions to ordinary differential equations (ODEs), emphasizing classification, techniques, and concepts associated with these methods. It discusses various solution strategies including collocation, Galerkin, and subdomain methods, along with derivations and examples. The goal is to minimize the residue in order to obtain algebraic equations that yield acceptable accuracy in the approximated solutions.
FEM: Introduction and Weighted Residual Methods
- 1. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Weighted Residual Methods Mohammad Tawfik
- 2. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • In this section we will be introduced to the general classification of approximate methods • Special attention will be paid for the weighted residual method • Derivation of a system of linear equations to approximate the solution of an ODE will be presented using different techniques
- 3. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Classification of Approximate Solutions of D.E.’s • Discrete Coordinate Method – Finite difference Methods – Stepwise integration methods • Euler method • Runge-Kutta methods • Etc… • Distributed Coordinate Method
- 4. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Distributed Coordinate Methods • Weighted Residual Methods – Interior Residual • Collocation • Galrekin • Finite Element – Boundary Residual • Boundary Element Method • Stationary Functional Methods – Reyligh-Ritz methods – Finite Element method
- 5. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Basic Concepts • A linear differential equation may be written in the form: xgxfL • Where L(.) is a linear differential operator. • An approximate solution maybe of the form: n i ii xaxf 1
- 6. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Basic Concepts • Applying the differential operator on the approximate solution, you get: 0 1 1 xgxLa xgxaLxgxfL n i ii n i ii xRxgxLa n i ii 1 Residue
- 7. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Handling the Residue • The weighted residual methods are all based on minimizing the value of the residue. • Since the residue can not be zero over the whole domain, different techniques were introduced.
- 8. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com General Weighted Residual Method
- 9. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Objective of WRM • As any other numerical method, the objective is to obtain of algebraic equations, that, when solved, produce a result with an acceptable accuracy. • If we are seeking the values of ai that would reduce the Residue (R(x)) allover the domain, we may integrate the residue over the domain and evaluate it!
- 10. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Evaluating the Residue xRxgxLa n i ii 1 xRxgxLaxLaxLa nn ...2211 n unknown variables 0 1 Domain n i ii Domain dxxgxLadxxR One equation!!!
- 11. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Using Weighting Functions • If you can select n different weighting functions, you will produce n equations! • You will end up with n equations in n variables. 0 1 Domain n i iij Domain j dxxgxLaxwdxxRxw
- 12. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Collocation Method • The idea behind the collocation method is similar to that behind the buttons of your shirt! • Assume a solution, then force the residue to be zero at the collocation points
- 13. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Collocation Method 0jxR 0 1 j n i jii j xFxLa xR
- 14. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example Problem
- 15. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com The bar tensile problem 0/ 00 ' 02 2 dxdulx ux sBC xF x u EA
- 16. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Bar application 02 2 xF x u EA n i ii xaxu 1 xRxF dx xd aEA n i i i 1 2 2 Applying the collocation method 0 1 2 2 j n i ji i xF dx xd aEA
- 17. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com In Matrix Form nnnnnn n n xF xF xF a a a kkk kkk kkk 2 1 2 1 21 22212 12111 ... ... ... Solve the above system for the “generalized coordinates” ai to get the solution for u(x) jxx i ij dx xd EAk 2 2
- 18. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Notes on the trial functions • They should be at least twice differentiable! • They should satisfy all boundary conditions! • Those are called the “Admissibility Conditions”.
- 19. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Using Admissible Functions • For a constant forcing function, F(x)=f • The strain at the free end of the bar should be zero (slope of displacement is zero). We may use: l x Sinx 2
- 20. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Using the function into the DE: • Since we only have one term in the series, we will select one collocation point! • The midpoint is a reasonable choice! l x Sin l EA dx xd EA 22 2 2 2 faSin l EA 1 2 42
- 21. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solving: • Then, the approximate solution for this problem is: • Which gives the maximum displacement to be: • And maximum strain to be: EA fl EA fl SinlEA f a 2 2 2 21 57.0 24 42 l x Sin EA fl xu 2 57.0 2 5.057.0 2 exact EA fl lu 0.19.00 exact EA lf ux
- 22. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com The Subdomain Method • The idea behind the subdomain method is to force the integral of the residue to be equal to zero on a subinterval of the domain
- 23. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com The Subdomain Method 0 1 j j x x dxxR 0 11 1 j j j j x x n i x x ii dxxgdxxLa
- 24. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Bar application 02 2 xF x u EA n i ii xaxu 1 xRxF dx xd aEA n i i i 1 2 2 Applying the subdomain method 11 1 2 2 j j j j x x n i x x i i dxxFdx dx xd aEA
- 25. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com In Matrix Form 11 2 2 j j j j x x i x x i dxxFadx dx xd EA Solve the above system for the “generalized coordinates” ai to get the solution for u(x)
- 26. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Using Admissible Functions • For a constant forcing function, F(x)=f • The strain at the free end of the bar should be zero (slope of displacement is zero). We may use: l x Sinx 2
- 27. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Using the function into the DE: • Since we only have one term in the series, we will select one subdomain! l x Sin l EA dx xd EA 22 2 2 2 ll fdxadx l x Sin l EA 0 1 0 2 22
- 28. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solving: • Then, the approximate solution for this problem is: • Which gives the maximum displacement to be: • And maximum strain to be: EA fl EA fl lEA fl a 22 1 637.0 2 2 l x Sin EA fl xu 2 637.0 2 5.0637.0 2 exact EA fl lu 0.10.10 exact EA lf ux fla l x Cos l EA l 1 0 22
- 29. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com The Galerkin Method • Galerkin suggested that the residue should be multiplied by a weighting function that is a part of the suggested solution then the integration is performed over the whole domain!!! • Actually, it turned out to be a VERY GOOD idea
- 30. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com The Galerkin Method 0Domain j dxxxR 0 1 Domain j n i Domain iji dxxgxdxxLxa
- 31. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Bar application 02 2 xF x u EA n i ii xaxu 1 xRxF dx xd aEA n i i i 1 2 2 Applying Galerkin method Domain j n i Domain i ji dxxFxdx dx xd xaEA 1 2 2
- 32. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com In Matrix Form Domain ji Domain i j dxxFxadx dx xd xEA 2 2 Solve the above system for the “generalized coordinates” ai to get the solution for u(x)
- 33. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Same conditions on the functions are applied • They should be at least twice differentiable! • They should satisfy all boundary conditions! • Let’s use the same function as in the collocation method: l x Sinx 2
- 34. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Substituting with the approximate solution: Domain j n i Domain i ji dxxFxdx dx xd xaEA 1 2 2 l l fdx l x Sin dx l x Sin l x Sina l EA 0 0 1 2 2 222 ll a l EA 2 22 1 2 EA fll EA f a 2 3 2 1 52.0 16
- 35. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Substituting with the approximate solution: (Int. by Parts) Domain j n i Domain i ji dxxFxdx dx xd xaEA 1 2 2 ll a l EA 2 22 1 2 EA fll EA f a 2 3 2 1 52.0 16 Domain ij l i j Domain i j dx dx xd dx xd dx xd x dx dx xd x 0 2 2 Zero!
- 36. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com What did we gain? • The functions are required to be less differentiable • Not all boundary conditions need to be satisfied • The matrix became symmetric!
- 37. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Summary • We may solve differential equations using a series of functions with different weights. • When those functions are used, Residue appears in the differential equation • The weights of the functions may be determined to minimize the residue by different techniques • One very important technique is the Galerkin method.