13b-Basic2Dshapes

# 13b-Basic2Dshapes - © William Klug MAE M168/CEE M135C...

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Unformatted text preview: © William Klug MAE M168/CEE M135C Introduction to Finite Element Methods Lecture 13b Basic Shape functions in 2-D © William Klug • Weak form SS Heat Transfer: • Requirements on – should be finite ⇒ T continuous – T = g on essential boundary • Minimal elements ∇ v i D ∇ T ( ) − Qv dV V ∫ + vq n dA S g ∫ + vhdA S n ∫ = subject to T = g on S g Summary and Review © William Klug Simplest Element in 2-D: Triangle • Interpolate a function T ( x , y ) among nodal values ( ) ( ) ( ) 1 1 1 2 2 2 3 3 3 , , , T T x y T T x y T T x y = = = © William Klug 3 Term Polynomial T x , y ( ) = a + bx + cy = 1 x y a b c 1 1 1 2 2 2 3 3 3 T a bx cy T a bx cy T a bx cy = + + = + + = + + ⇒ T 1 T 2 T 3 T e = 1 x 1 y 1 1 x 2 y 2 1 x 3 y 3 C a b c α • 3 nodal values, need 3 term polynomials © William Klug Shape Functions α = C − 1 T e ⇒ T x , y ( ) = 1 x y C − 1 N x , y ( ) T 1 T 2 T 3 ( ) [ ] , 1 a T x y x y b c = T 1 T 2 T 3 T e = 1 x 1 y 1 1 x 2 y 2 1 x 3 y 3 C a b c α © William Klug Shape Functions ( ) [ ] ( ) ( ) ( ) 1 1 2 3 , 1 , , , x y x y N x y N x y N x y − = = N C 1 1 2 2 3 3 1 1...
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## This note was uploaded on 08/09/2011 for the course MAE 168 taught by Professor Klug during the Spring '11 term at UCLA.

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13b-Basic2Dshapes - © William Klug MAE M168/CEE M135C...

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