TriangleElement_Seepage - CVEN4304 Structural Analysis...

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Unformatted text preview: CVEN4304 Structural Analysis & Finite Elements Triangle Element Chongmin Song School of Civil and Environmental Engineering University of New South Wales Introduction Triangle elements are the most versatile in mesh generation. Description of mesh (as in 1D case) Global nodal number Element number Local nodal number ( counter-clockwise ) Element connectivity table Nodal functions One element Nodal hydraulic head: h ( e ) 1 , h ( e ) 2 , h ( e ) 3 Nodal flux: Q ( e ) 1 , Q ( e ) 2 , Q ( e ) 3 Shape function Linear interpolation of the nodal values h ( x , z ) = a + a 1 x + a 2 z At the 3 nodes of element 1 x ( e ) 1 z ( e ) 1 1 x ( e ) 2 z ( e ) 2 1 x ( e ) 3 z ( e ) 3 a a 1 a 2 = h ( e ) 1 h ( e ) 2 h ( e ) 3 Determinant of the coefficient matrix 2 A ( e ) = ( x ( e ) 2 z ( e ) 3- x ( e ) 3 z ( e ) 2 ) + ( x ( e ) 3 z ( e ) 1- x ( e ) 1 z ( e ) 3 ) + ( x ( e ) 1 z ( e ) 2- x ( e ) 2 z ( e ) 1 ) The solution for constants a , a 1 and a 2 a = 1 2 A ( e ) (( x ( e ) 2 z ( e ) 3- x ( e ) 3 z ( e ) 2 ) h ( e ) 1 + ( x ( e ) 3 z ( e ) 1- x ( e ) 1 z ( e ) 3 ) h ( e ) 2 + ( x ( e ) 1 z ( e ) 2- x ( e ) 2 z ( e ) 1 ) h ( e ) 3 ) a 1 = 1 2 A ( e ) (( z ( e ) 2- z ( e ) 3 ) h ( e ) 1 + ( z ( e ) 3- z ( e ) 1 ) h ( e ) 2 + ( z ( e ) 1- z ( e ) 2 ) h ( e ) 3 ) a 2 = 1 2 A ( e ) (( x ( e ) 3- x ( e ) 2 ) h ( e ) 1 + ( x ( e ) 1- x ( e ) 3 ) h ( e ) 2 + ( x ( e ) 2- x ( e ) 1 ) h ( e ) 3 ) Shape function (conted) Hydraulic head within element h ( x , z ) = N ( e ) 1 ( x , z ) h ( e ) 1 + N ( e ) 2 ( x , z ) h ( e ) 2 + N ( e ) 3 ( x , z ) h ( e ) 3 = [ N ( e ) ( x , z )] { h ( e ) } Shape functions [ N ( e ) ( x , z )] = [ N ( e ) 1 ( x , z ) N ( e ) 2 ( x , z ) N ( e ) 3 ( x , z ) ] where N ( e ) 1 ( x , z...
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TriangleElement_Seepage - CVEN4304 Structural Analysis...

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