me180_hw04

# me180_hw04 - element • You are to plot the solution(nodal...

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PROJECT 4: “POTENTIALS AND EFFICIENT SOLUTION TECHNIQUES” Solve the following boundary value problem, with domain Ω = (0 ,L ), analytically: d dx ± A 1 ( x ) du dx ² = k 2 sin ( 2 πkx L ) A 1 ( x ) = 10 DIFFERENT SEGMENTS ( SEE BELOW ) k = 22 , L = 1 , u (0) = 0 . 2 , u ( L ) = - 0 . 1 (1) For A 1 FOR 0 . 0 < x < 0 . 1 A 1 = 2 . 5 FOR 0 . 1 < x < 0 . 2 A 1 = 2 . 0 FOR 0 . 2 < x < 0 . 3 A 1 = 1 . 75 FOR 0 . 3 < x < 0 . 4 A 1 = 0 . 5 FOR 0 . 4 < x < 0 . 5 A 1 = 1 . 75 FOR 0 . 5 < x < 0 . 6 A 1 = 2 . 75 FOR 0 . 6 < x < 0 . 7 A 1 = 0 . 25 FOR 0 . 7 < x < 0 . 8 A 1 = 1 . 75 FOR 0 . 8 < x < 0 . 9 A 1 = 3 . 0 FOR 0 . 9 < x < 1 . 0 A 1 = 1 . 5 (2) Solve this with the ﬁnite element method using linear equal-sized ele- ments. Use 100, 1000 and 10000 elements. You are to write a Precon- ditioned Conjugate-Gradient solver. Use the diagonal preconditioning given in the notes. The data storage is to be element by element (sym- metric) and the matrix vector multiplication is to be done element by

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Unformatted text preview: element. • You are to plot the solution (nodal values) for each N . • You are to plot 1 e N def = || u-u N || A 1 (Ω) || u || A 1 (Ω) , || u || A 1 (Ω) def = s Z Ω du dx A 1 du dx dx, (3) for each N . • You are to plot POTENTIAL ENERGY = J ( u N ) (4) for each N . • You are to plot the number of PCG-solver iterations for each N for a stopping tolerance of 0 . 000001. • Use a Gauss integration rule of level 5. • Check your Conjugate Gradient generated results against a regular Gaussian solver, for example the one available in Matlab. 2...
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me180_hw04 - element • You are to plot the solution(nodal...

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