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8a-Linear-shp-fns-2010s

8a-Linear-shp-fns-2010s - MAE M168/CEE M135C Introduction...

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© William Klug MAE M168/CEE M135C Introduction to Finite Element Methods Lecture 8a 1-D FEM: Interpolation & Linear Shape Functions
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© William Klug FEM in 1-D Recall basic procedure: 1. Discretize ( ) q x x , u ( x ) (1) (2) (3) (4) 1 2 3 4 5
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© William Klug Linear Interpolation 2. Approximate local behavior using DOF Define by local polynomial interpolation of Let be nodal positions. Define: Nodal Displacements ( ) I u x ( ). u x ( x 1 , D 1 ) ( x 2 , D 2 ) ( x 3 , D 3 ) ( x 4 , D 4 ) ( x 5 , D 5 ) x x 1 x 2 x 3 x 4 x 5 u ( x ) ( x 1 , D 1 ) ( x 2 , D 2 ) ( x 3 , D 3 ) ( x 4 , D 4 ) ( x 5 , D 5 ) x x 1 x 2 x 3 x 4 x 5 ( ) I u x 1 2 5 , , ..... , x x x ( ) 1,2, ..... i i D u x i = =
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© William Klug Linear Interpolation Interpolation: Connecting the dots, to define in between the D i ’s. Use interpolation as approximation in R-R or Galerkin. ( x 1 , D 1 ) ( x 2 , D 2 ) ( x 3 , D 3 ) ( x 4 , D 4 ) ( x 5 , D 5 ) x x 1 x 2 x 3 x 4 x 5 ( ) I u x ( ) I u x ( ) ( ) I u x u x =
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© William Klug Interpolation Requirements Derivatives in weak form must not blow up (finite energy) must be continuous.
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