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With
Galerkin's (1915) method
or the
Bubnov
8
(1913)-Galerkin method
, these
mp
equations are
used to find the
mp
unknown coefficients
Often, the
N
ui
are
chosen to be members of a complete set of functions. The trial solutions are chosen to satisfy
all boundary conditions.
Since Galerkin's method, the most popular of the weighted-residual techniques, will be
applied extensively to solids in Section 7.4, no examples will be treated here.
u
i
,i
=
1
,
2
,
...
,mp,
in
u
(
x
).
7.3.7
Method of Moments
In what is called the
method of moments
[Yamada, 1950], Eq. (7.26) is used with
Ψ
j
,j
=
1
,
2
,
,n
selected as the first
n
members of a complete set of functions. Such series as
an ordinary polynomial, trigonometric, and Tchebychev polynomial can be complete. If
an ordinary polynomial is used, then
Ψ
j
...
=
x
j
−
1
,j
=
...
.
Successively higher
“moments” of the residual are required to be zero. Note that for the first approximation,
i.e.,
Ψ
1
1
,
2
,
,n
1
,
the method of moments is the same as the subdomain method with the
subdomain equal to the whole domain. The method of moments is sometimes referred to
as the
integral method
of von Karman
9
=
(1921) and Pohlhausen
10
(1921).
7.3.8
Least Squares
In Section 7.3.2, the least squares method was used to minimize the sum of the squares of
the residuals at some selected points. In this section, the integral form of the least squares
method is considered. Here, the integral (over domain
V
) of the weighted square of the
residual is required to be a minimum. That is,
R
2
dV
is minimized
(7.29)
V
Choose
W
j
(
to be positive, so that the integrand is positive. Substitution of Eq. (7.9) into
Eq. (7.29) gives an integral containing
x
)
u
as unknowns. For a
p
-dimensional residual vector
R
write Eq. (7.29) in matrix form, as
LN
u
f
T
W
LN
u
f
dV
R
T
WR
dV
=
u
−
u
−
V
V
f
T
Wf
dV
u
T
T
WLN
u
u
T
T
Wf
=
(
)
−
(
)
+
LN
u
u
2
LN
u
V
=
minimum
(7.30)
where
R
is a
pm
1 vector and
W
is a diagonal weighting matrix with positive elements.
The necessary condition that the minimum in Eq (7.30) be achieved is found by setting the
×
8
I.G. Bubnov was a Russian shipbuilding engineer. In 1913 he published the fundamentals of what is known today
as the Bubnov-Galerkin Method. Galerkin generalized the method in 1915.
9
Theodore von Karman (1881-1963) was born in Hungary and received his PhD from the University of G ottingen,
Germany, in 1908. He moved to the United States in 1930 and became director of the Guggenheim Aeronautical Lab
and the Jet Propulsion Lab. In 1924 he co-authored the topic
General Aerodynamic Theory
and in 1935 he developed
the theory of supersonic drag, now called the Karman vortex trail. He contributed significantly to thermodynamics,
aerodynamics, and hydrodynamics and is recognized for his pioneering efforts in the development of high speed
aircraft.
10
Karl Pohlhausen was a German scientist who studied at the University of G ottingen school of applied mechanics.
He was a student of Prandtl and, after World War I, followed von Karman from G ottingen to Aachen. He moved
to Wright Field in the United States after World War II and worked in fluid mechanics.
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