Model2DEqn_Ch8

Model2DEqn_Ch8 - FINITE ELEMENT FORMULATION OF POISSONS...

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x y Γ Γ Ω e e α ds n ˆ Domain of the model problem Finite element mesh of the domain A typical finite element with its boundary FINITE ELEMENT FORMULATION OF POISSON’S EQUATION IN TWO DIMENSIONS 12 21 00 Poisson's Equation is given by ( 0 in the model equation of the book) aaa === 11 22 11 22 (, )i n where , , and are known functions of and . uu aa f x y xx y y f x y Ω ⎛⎞ ∂∂ −− = ⎜⎟ ⎝⎠
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STEP-BY-STEP FORMULATION OF THE PROBLEM 1. Model Differential Equation 2. Weak Form 3. Finite Element Model x μ a 11 u x y μ a 22 u y = f ( x, y )( 1 ) 0= Z e μ a 11 w x u x + a 22 w y u y wf dxdy I Γ e wq n ds (2 a ) q n = a 11 u x n x + a 22 u y n y (2 b ) u ( x,y ) n X j =1 u j ψ e j ( 3 ) [ K e ] { u e } = { f e } + { Q e } (4) K e ij = Z e μ a 11 ∂ψ e i x e j x + a 22 e i y e j y dxdy f e i = Z e ψ e i fdxdy Q e i = I Γ e ψ e i q n ds (5)
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Finite Element Approximation of the Solution u ( x,y ) u h ( x,y ) x x y y Ω u ( x,y ) 12 3 11 22 33 (, ) ee e e h uxy c cxcy uuu ψψψ =+ + =++ e Ω Ω e u 1 e u 2 e u 3 1 2 3 1 n e hj j j uxy u xy ψ = = Finite Element Approximation of the Solution: where n denotes the number of nodes in the element. 3, linear triangular element 4, linear rectantangular element 6, quadratic triangular element 8 or 9 quadratic rectangular element n =
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y x 1 3 2 1 1 3 2 ψ 1 1 1 3 2 ψ 2 1 1 3 2 ψ 3 y x 1 4 3 2 b a y _ x _ 1 1 4 3 2 ψ 1 1 1 4 3 2 ψ 2 1 1 4 3 2 ψ 3 1 1 4 3 2 ψ 4 Linear Triangular Element Linear Rectangular Element 1 (, ) ( ) 2 , , 2 2 triangle area () ii i i ij k k j k k xy x y A yy A xx ψα β γ α =+ + =− = × 12 34 The general form of is (, ) 1 1 , 1 , , i i i i x y x y x yx y ab x y a b ψ λ ψψ + + ⎛⎞ = ⎜⎟ ⎝⎠ ==
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For element-wise constant values of a e 11 ,a e 22 ,and f e 0 ,w e have K e ij = Z e μ a 11 ∂ψ e i x e j x + a 22 e i y e j y dxdy = a e 11 Z e e i x e j
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Model2DEqn_Ch8 - FINITE ELEMENT FORMULATION OF POISSONS...

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