18. 2-D_Laplace_Equation_Example

18. 2-D_Laplace_Equation_Example - 5.1 2-D Laplace Equation...

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1 of 4 5.1 2-D Laplace Equation Example ± A long, square bar 8 cm by 8 cm cross-section ± Two sides held at 100ºC, two sides at 0ºC ± Find the steady state temperature ( ) y x T , across the bar cross-section Governing equations for () y x T , : 2-D Laplace: (1) 0 2 2 2 2 = + y T x T (2) given BCs Choose Δ x = 2 cm Δ y = 2 cm nine unknown T values. Equation (1) holds at arbitrary ( ) i i y x , Substitute second order central difference 0 2 2 2 1 , , 1 , 2 , 1 , , 1 = Δ + + Δ + + + y T T T x T T T j i j i j i j i j i j i 4 1 , 1 , , 1 , 1 , + + + + + = j i j i j i j i j i T T T T T ( À ) Discretized P.D.E. ± Apply equation ( À ) at each of the nine interior locations ± Apply BCs on edges ± A set of nine linear equations for nine unknown T ’s 100 100 0 0
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2 of 4 Dicretized P.D.E. Gauss-Seidel Iteration at 1 1 , y x 4 1 , 1 = T (1) 4 1 , 1 = new T at 2 1 , y x 4 (2) 4 at 3 1 , y x 4 (3) 4 at 1 2 , y x 4 (4) 4 at 2 2 , y x 4 2 , 2 = T (5) 4 2 , 2 = new T at 3 2 , y x 4 (6) 4 at 1 3 , y x 4 (7) 4 at 2 3 , y x 4 2 , 3 = T (8) 4 2 , 3 = new T at 3 3 , y x 4 (9) 4 The discretized P.D.E. + BCs lead to a set of nine linear equations to solve for nine unknowns.
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18. 2-D_Laplace_Equation_Example - 5.1 2-D Laplace Equation...

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