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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 Fall '09
 Trigraph, Second moment of area, 1 L, 2 L, Kikuchi, mε

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