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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: C C 1 2 3 4 x C y C xy + + + 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 U V t x y ψ ψ ψ + + = with Pagoda functions. Define averaging operators and x y A A ( ) ( ) , 1, , 1, , , 1 , , 1 , 2 , 2 , 1 4 6 1 4 6 The advection equation becomes 0 x i j i j i j i j y i j i j i j i j i j x y y x i j x y i j A a a a a A a a a a da A A UA a VA a dt δ δ + + = + + = + + + + = 1
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where 1, 1, 2 , , 1 , 1 2 , 2 2 i j i j x i j i j i j y i j a a a x a a a y δ δ + + = = The product of averaging operators couples the time derivatives at 9 different nodes and generates a band matrix with a wide band width. x y A A An efficient solution can be done by: 1. Evaluate the spatial derivatives at every node point first by evaluating the tri-diagonal system: 2 , , 2 , , 1 1 1, , 1, for and . These are the independent "compact scheme" equations, i.e.
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