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me180_quiz03_solutions

me180_quiz03_solutions - University of California Berkeley...

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University of California, Berkeley ME 180, Engineering Analysis Using the Finite Element Method Spring 2008 Instructor: T. Zohdi Quiz 3 Solutions Problem 1 Calculate the sti ff ness and force matrices corresponding to the problem d dx c ( x + 1) 2 du dx ! + 3 u = 4 e - x in Ω = (0 , L ) , u = 37 on Γ u = L , du dx = - 11 on Γ q = 0 , with a 1D mesh of 5 elements of arbitrary size. Solution: (15 points) The first step will be to derive the weak form. Since you should be familiar with this by now, my derivation will be abbreviated: Z Ω wR d Ω = 0 , Z Ω w " d dx c ( x + 1) 2 du dx ! + 3 u - 4 e - x # d Ω = 0 , Z Ω " d dx wc ( x + 1) 2 du dx ! - dw dx c ( x + 1) 2 du dx # + 3 wu - 4 we - x d Ω = 0 , Z Ω - c ( x + 1) 2 dw dx du dx + 3 wu - 4 we - x d Ω + Z Γ q wc ( x + 1) 2 du dx n x d Γ = 0 , Z Ω " - c ( x + 1) 2 dw dx du dx + 3 wu - 4 we - x # d Ω + 11 cw (0) = 0 . Now, we recall that the solution and the weighing function over an element e are approximated using the same interpolation functions, as u = N e I u e I , w = N e I w e I , where the N e I are the element interpolation functions, and the u e I and w e I are the nodal values of the solution and the weighing function for the element, respectively. Note that, since we don’t have to do any calculations, we will ignore the isoparametric formulation, and just assume that N e I = N e I ( x ).
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