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 • Pascal’s 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 α α α α = = + + = + + + © 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 ξ ξ 1 1 − Standard Domain (Natural) j X i X X Actual Domain (Physical) ( ) ( ) ( ) ( ) 1 1 2 2 X N X N X ξ ξ ξ ξ = + = 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 δ = ( ) ( )...
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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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14a-2D-Isoparametric - © William Klug MAE M168/CEE M135C...

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