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TriangleElement_Seepage - CVEN4304 Structural Analysis...

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CVEN4304 Structural Analysis & Finite Elements Triangle Element Chongmin Song School of Civil and Environmental Engineering University of New South Wales
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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
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Shape function Linear interpolation of the nodal values h ( x , z ) = a 0 + 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 0 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 0 , a 1 and a 2 a 0 = 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 )
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Shape function (cont’ed) 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 ) = 1 2 A ( e ) (( x ( e ) 2 z ( e ) 3 - x ( e ) 3 z ( e ) 2 ) + ( z ( e ) 2 - z ( e ) 3 ) x + ( x ( e ) 3 - x ( e ) 2 ) z ) N ( e ) 2 ( x , z ) = 1 2 A ( e ) (( x ( e ) 3 z ( e ) 1 - x ( e ) 1 z ( e ) 3 ) + ( z ( e ) 3 - z ( e ) 1 ) x + ( x ( e ) 1 - x ( e ) 3 ) z ) N ( e ) 3 ( x , z ) = 1 2 A ( e ) (( x ( e ) 1 z ( e ) 2 - x ( e ) 2 z ( e ) 1 ) + ( z ( e ) 1 - z ( e ) 2 ) x + ( x ( e ) 2 - x ( e ) 1 ) z ) Note that the area of an element A ( e ) must not be zero!
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Shape function (cont’ed)
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