Information Technology Reference
In-Depth Information
Clearly, this Hermitian polynomial leads to the exact deflection.
Comparison of Galerkin's Solution
Deflection
Moment
Shear Force
w
=
β
p
0
L
2
100
EI
M
=
β
p
0
L
2
V
=
β
p
0
L/
10
β
for
ξ
=0
.
5
β
for
ξ
=0
β
for
ξ
=0
.
5
β
for
ξ
=1
Exact Soln. A
0.234375
−
0.0666
0.029166
−
1
−
−
.
Method B
0.231481
0.05555
0.027777
1
666
−
−
Methods C and D
0.234375
0.0666
0.029166
1
1 case, the error tends to grow in moving from the deflection to the
moment to the shear force.
Note that for the
m
=
7.4.3 Kantorovitch's Method
A variation of the Ritz method is the
Kantorovitch
12
method
or the
method of lines
which can
be applied to reduce the partial differential equations of boundary value problems to the
solution of lower dimensional (even one-dimensional) boundary value problems. In this
one-dimensional case, the method involves the solution of ordinary differential equations.
The method still begins with the trial solution
=
i
=
1
N
ui
u
i
are func-
tions of one of the independent variables. In the case of a two-dimensional problem, a
possible trial solution is
u
u
i
,
but now the
m
u
(
x, y
)
=
N
ui
(
y
)
u
i
(
x
)
(7.66)
=
i
1
The unknown
of Eq. (7.66) are determined so that a variational principle is satisfied.
Instead of the system of linear algebraic equations of the Ritz method, we obtain a boundary
value problem of a system of ordinary differential equations for the unknown function
u
i
(
x
)
u
i
(
In setting up the problem, it is not necessary to choose
N
ui
to satisfy the
x
boundary
conditions, since they may be prescribed as boundary conditions for the functions
x
).
u
i
.
EXAMPLE 7.8 Kantorovitch's Method
Consider the torsion of a prismatic bar with the Prandtl stress function defined as
φ
∂ψ
∂
φ
∂ψ
∂
τ
=
G
and
τ
=−
G
(1)
xy
xz
z
y
Note that the definition of
ψ
differs from that in Chapter 1, Eq. (1.155) in that the constant
φ
G
is included. This definition will simplify the compatibility condition. Equation (1.159)
12
Leonid Vitaljevich Kantorovitch (1912-1986) was born in Russia, educated in Leningrad, and worked in Siberia.
He entered Leningrad University in the school of mechanics and mathematics at the age of 14 and was appointed
as a full professor while still a teenager. By the age of 18, when he received a degree, he had published 11 scientific
papers. In 1939 he published “Mathematical Methods of Organization and Planning of Production” which outlined
the simplex method of linear programming, an important technique that until recently was thought to have been
developed in the West several years later. The hot and cold war stifled the appreciation of much of his work. His
achievements in functional analysis and approximate methods are widely recognized. Probably his contributions
to the foundations of modern economic theory are his most important accomplishments. He was awarded a Nobel
Prize in 1975.
Search WWH ::
Custom Search