CFD lecture 12 - Computational Fluid Dynamics Lecture 12...

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Computational Fluid Dynamics Lecture 12 Finite Elements in 2-D Straightforward extension of 1-D ideas Simplest FE is rectangle (also consider triangles) 1. Bilinear interpolation function: CC 12 3 4 xC yC x y + ++ Coefficients determined by function values at four vertices. Reduce to linear interpolation along x-y axis. The product of Chapeau functions in x and y called “Pagoda” functions. Unity at central node. And vanishing to zero at 8 neighboring nodes. Consider 2-D advection equation: 0 UV txy ψ ψψ ∂∂∂ + += with Pagoda functions. Define averaging operators and xy AA () ,1 , , 1 , ,, 1 , , 1 , 2, 1 4 6 1 4 6 The advection equation becomes 0 xi j i j i j i j yi j i j i j i j ij y x i j x y i j Aa a a a a a a da UA a VA a dt δδ +− =+ + + ++= 1
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where 1, 1, 2, ,1 2 2 ij xi j yi j aa a x a y δ +− = = The product of averaging operators couples the time derivatives at 9 different nodes and generates a band matrix with a wide band width. xy AA An efficient solution can be done by: 1. Evaluate the spatial derivatives at every node point first by evaluating the tri-diagonal system: , , 11 1, , 1, for and . These are the independent "compact scheme" equations, i.e.
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This note was uploaded on 06/08/2011 for the course EOC 6850 taught by Professor Slinn during the Spring '07 term at University of Florida.

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CFD lecture 12 - Computational Fluid Dynamics Lecture 12...

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