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What are weighted residual methods?

How to apply Galerkin Method to the finite element model?

#WikiCourses #Num001

https://wikicourses.wikispaces.com/TopicX+Approximate+Methods+-+Weighted+Residual+Methods

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- 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.

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