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Unformatted text preview: 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 = (0 , L ) , u = 37 on u = L , du dx =- 11 on q = , 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 wRd = , Z w " d dx c ( x + 1) 2 du dx ! + 3 u- 4 e- x # d = , Z " d dx wc ( x + 1) 2 du dx !- dw dx c ( x + 1) 2 du dx # + 3 wu- 4 we- x d = , Z - c ( x + 1) 2 dw dx du dx + 3 wu- 4 we- x d + Z q wc ( x + 1) 2 du dx n x d = , Z "- c ( x + 1) 2 dw dx du dx + 3 wu- 4 we- x # d + 11 cw (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 dontthe solution and the weighing function for the element, respectively....
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This note was uploaded on 08/01/2008 for the course ME 180 taught by Professor Zohdi during the Spring '08 term at University of California, Berkeley.
- Spring '08