11a-Bar-FEM-2010s

11a-Bar-FEM-2010s - MAE M168/CEE M135C Introduction to...

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© William Klug MAE M168/CEE M135C Introduction to Finite Element Methods Lecture 10a 1-D Elastic Bar FEM

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© William Klug Summary & Review Piecewise Linear interpolation Piecewise Quadratic interpolation Higher-order… Global approximation i j N i (x) N j (x) x N(x) ( ) ( ) ( ) i i j j u x N x u N x u = + ( 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 1 2 2 1 ( ) ( ) ( ) ( ) N i i i u X D N X D N X D N 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 piecewise quadratic segments ( ) ( ) ( ) ( ) i i j j k k u x N x u N x u N x u = + + x 1 j k i l l 1 N 2 N 3 N 4 N 5 N 1 N 2 N 3 N 4 N 5 N
© William Klug Summary and Review j i 0 X L Global (physical) x j i 0 l Local (physical) ξ j i 0 + 1 1 Natural (unitless) X = X i + x N i = 1 x l = 1 2 1 ( ) 0 i j 1 1 ( ) i N 1 i j 1 1 ( ) j N 1 0 N j = x l = 1 2 1 + ( ) ( ) ( ) 2 2 1 1 2 2 i l N x x l l ξξ = = ( )( ) 2 2 1 1 2 j l N x x l = = + ( ) ( ) 2 4 1 1 2 k N x x l l = = + ( ) j N i j k 1 1 0 N i ( ) i j k 1 1 0 N k ( ) i j k 1 1 0 x = l 2 + 1 ( )

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© William Klug Discretize into elements Interpolation with piece X ise polynomial basis (shape) functions Plug approximation into weak form Galerkin approach to FEM 1 1 2 2 1 ( ) ( ) ( ) ( ) N i i i u X D N X D N X D N 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 ( 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 piecewise quadratic segments 1 2 3 4 5 1 N 2 N 3 N 4 N 5 N (1) (2) 1 N 2 N 3 N 4 N 5 N (1) (2) (3) (4) 1 2 3 4 5