14a-2D-Isoparametric

14a-2D-Isoparametric - 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 14a Isoparametric Elements in 2-D William Klug Simple 2-D elements Pascals Triangle Summary and Review T x , y ( ) = a + bx + cy = T 1 N 1 + T 2 N 2 + T 3 N 3 Linear Triangle Bi-linear rectangle 1 2 3 4 ( )( ) u u a bx c dy x y xy = = + + = + + + 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 + vhdA S q = subject to T = g on S T Summary and Review S q S T William Klug Motivation for Isoparametric Elements Deal with distorted shapes. Compute integrals over standard domain. X Y 1 2 3 4 v i D T ( ) Qv dV V + vhdA S q = subject to T = g on S T William Klug Isoparametric Mapping in 1-D ( ) X 1 1 Standard Domain (Natural) j X i X X Actual Domain (Physical) ( ) ( ) ( ) ( ) 1 1 2 2 X N X N X = + = N X Transformation interpolates between nodal positions in actual physical domain William Klug Isoparametric Mapping in 2-D ( ) ( ) ( ) ( ) 3 1 3 1 , , , , i i i i i i X r s N r s X Y r s N r s Y = = = = r s 2 1 3 1 1 X Y Standard Domain (Natural) Actual Domain (Physical) William Klug Basic Idea Use shape functions to create mapping between actual and natural coordinate systems. Build shape functions and do integrals in natural coordinates . William Klug 3-node Triangle in Natural Coordinates Building shape functions in ( r , s ) Linear in r & s ( ) , i j j ij N r s = ( ) ( )...
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14a-2D-Isoparametric - William Klug MAE M168/CEE M135C...

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