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- 1. Numerical Solution of Boundary Value Problems Weighted Residual Methods ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 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 ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 1
- 2. 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 ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 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 ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 2
- 3. Basic Concepts • A linear differential equation may be written in the form: L( f ( x )) = g ( x ) • Where L(.) is a linear differential operator. • An approximate solution maybe of the form: n f (x ) = ∑ aiψ i (x ) i =1 ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik Basic Concepts • Applying the differential operator on the approximate solution, you get: ⎛ n ⎞ L( f ( x )) − g ( x ) = L⎜ ∑ aiψ i (x )⎟ − g (x ) ⎝ i =1 ⎠ n = ∑ ai L(ψ i ( x )) − g (x ) ≠ 0 i =1 n ∑ a L(ψ (x )) −g (x ) = R(x ) i =1 i i Residue ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 3
- 4. 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. ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik General Weighted Residual Method ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 4
- 5. 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! ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik Evaluating the Residue n ∑ a L(ψ (x )) −g (x ) = R(x ) i =1 i i a1 L(ψ 1 ( x )) + a2 L(ψ 2 ( x )) + ... + an L(ψ n ( x )) − g ( x ) = R( x ) One equation!!! n unknown variables ⎛ n ⎞ ∫ R(x )dx = Domain ⎜ ∑ ai L(ψ i (x )) − g (x )⎟dx = 0 Domain ∫ ⎝ i =1 ⎠ ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 5
- 6. Using Weighting Functions ⎛ n ⎞ ∫ w j (x )R(x )dx = Domain w j (x )⎜ ∑ ai L(ψ i (x )) − g (x )⎟dx = 0 Domain ∫ ⎝ i =1 ⎠ • If you can select n different weighting functions, you will produce n equations! • You will end up with n equations in n variables. ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 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 ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 6
- 7. Collocation Method R(x j ) = 0 R (x j ) = ∑ a L(ψ (x )) − F (x ) = 0 n i i j j i =1 ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik Example Problem ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 7
- 8. The bar tensile problem ∂ 2u EA 2 + F ( x ) = 0 ∂x BC ' s x=0⇒u =0 x = l ⇒ du / dx = 0 ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik Bar application ∂ 2u EA 2 + F (x ) = 0 ∂x n u ( x ) = ∑ aiψ i ( x ) i =1 n d 2ψ i ( x ) EA∑ ai + F (x ) = R(x ) Applying the collocation method i =1 dx 2 d 2ψ i (x j ) + F (x j ) = 0 n EA∑ ai i =1 dx 2 ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 8
- 9. In Matrix Form ⎡ k11 k 21 ... k n1 ⎤ ⎧ a1 ⎫ ⎧ F ( x1 )⎫ ⎢k ⎥ ⎪a ⎪ ⎪ F ( x )⎪ ⎢ 12 k 22 ... k n 2 ⎥ ⎪ 2 ⎪ ⎪ 2 ⎪ d 2ψ i ( x ) ⎨ ⎬ = −⎨ ⎬ kij = EA ⎢ M M O M ⎥⎪ M ⎪ ⎪ M ⎪ dx 2 x = x ⎢ ⎥ j ⎣k1n k2n ... k nn ⎦ ⎪an ⎪ ⎩ ⎭ ⎪ F ( xn )⎪ ⎩ ⎭ Solve the above system for the “generalized coordinates” ai to get the solution for u(x) ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik Notes on the trial functions • They should be at least twice differentiable! • They should satisfy all boundary conditions! • Those are called the “Admissibility Conditions”. ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 9
- 10. 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: ⎛ πx ⎞ ψ (x ) = Sin⎜ ⎟ ⎝ 2l ⎠ ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik Using the function into the DE: d 2ψ (x ) ⎛π ⎞ ⎛ πx ⎞ 2 EA = − EA⎜ ⎟ Sin⎜ ⎟ ⎝ 2l ⎠ ⎝ 2l ⎠ 2 dx • Since we only have one term in the series, we will select one collocation point! • The midpoint is a reasonable choice! ⎡ ⎛π ⎞ 2 ⎛ π ⎞⎤ ⎢− EA⎜ ⎟ Sin⎜ ⎟⎥{a1} = −{ f } ⎢ ⎣ ⎝ 2l ⎠ ⎝ 4 ⎠⎥ ⎦ ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 10
- 11. Solving: f 4 2l 2 f l2 f a1 = = ≈ 0.57 EA(π 2l ) Sin(π 4) π 2 EA 2 EA • Then, the approximate l2 f ⎛ πx ⎞ u ( x ) ≈ 0.57 Sin⎜ ⎟ solution for this problem is: EA ⎝ 2l ⎠ • Which gives the maximum l2 f displacement to be: u (l ) ≈ 0.57 (exact = 0.5) EA • And maximum strain to be: u x (0) ≈ 0.9 (exact = 1.0) lf EA ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik Homework #11 • Solve the beam bending problem, for beam displacement, for a simply supported beam with a load placed at the center of the beam using – Any weighting function d 4w – Collocation Method = F ( x) • Use three term Sin series dx 4 that satisfies all BC’s w(0) = w(l ) = 0 • Write a program that d 2 w(0) d 2 w(l ) produces the results for = =0 n-term solution. dx 2 dx 2 ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 11
- 12. Exact Solution x 3 13 x w( x) = + 0 < x < 1/ 2 12 60 x3 x 2 7 x 3 =− + − + 1/ 2 < x < 1 12 4 15 10 ENME 602 Spring 2007 Dr. Eng. Mohammad Tawfik 12

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