hw4_98W - HW#2 1 Homework#4 AM 502 Differential Equation...

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Homework #4 AM 502 Differential Equation Methods in Mechanics 1998 Winter, Kikuchi 1. Consider an oscillatory coefficient shown in below 0.1 0.2 0.3 0.4 x a(x) 0.5 1.0 1.5 and the functional F u ( 29 = 1 2 a x ( 29 du dx 2 + sin 2 π x ( 29 u 2 dx 0 1 - 1 + x 2 ( 29 udx 0 1 for the minimization problem min u F u ( 29 among the admissible functions such that u 0 ( 29 = u 1 ( 29 = 0 . (1) Find the first variation δ F of the functional F and its Euler’s equation by setting δ F = 0 for every δ u = 0. (2) Find the homogenized coefficient a H and homogenized functional F H u H ( 29 , and then solve the homogenized problem to find u H that minimizes the homogenized functional F H u H ( 29 . Furthermore, estimate the maximum difference of u H - u and d dx u - u H ( 29 based on the homogenization asymptotic expansion. (3) Solve the original problem by FEM or FDM, and compare these results with the one obtained by the homogenization method. 2.
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hw4_98W - HW#2 1 Homework#4 AM 502 Differential Equation...

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