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7.2
Trial Functions
Suppose the governing differential equations of motion subject to prescribed boundary
conditions are to be solved for the single dependent variable
u
or the vector
u
. For example,
u
]
T
for three-dimensional solids. For the methods of concern here,
u
is approximated by trial solutions
=
[
u
1
u
2
u
3
]
T
=
vw
[
u
u
of the familiar form
m
(
)
(
)
=
1
(
)
u
(
x
)
=
N
u
x
u
or
u
x
u
i
N
ui
x
(7.9a)
i
=
for a scalar
u
and
(
)
=
(
)
u
(7.9b)
for a vector
u
, where
N
ui
are linearly independent chosen functions called
trial, basis,
or
shape
functions
. As indicated in previous chapters,
N
u
(
u
x
N
u
x
x
)
is a matrix (or row vector, if appropriate)
u
m
]
T
are unknown or free parameters (scalar values,
and in some cases functions) that are to be determined in some good “fit” sense. If
u
and
formed of
N
ui
,
and
u
=
[
u
0
u
1
u
2
u
3
...
u
are
p
×
1 vectors and
u
is a
pm
×
1 vector, then
N
u
(
x
)
is a
p
×
pm
matrix. For each
r,
1
≤
r
≤
p,
the
r
th row of
N
u
(
x
)
has
m
contiguous non-zero elements, the first of which
is the
th entry in the row. The trial functions should be chosen such that the
approximation improves as the number of terms in the solution increases. As discussed
in Section 7.3.9, convergence can be defined in terms of
(
mr
−
m
+
1
)
. If any desired
accuracy can be obtained by simply increasing the number of terms in the linear sum of
Eq. (7.9a), the set of functions
N
ui
is said to be
complete
. For convergence, it is usually
essential that the
N
ui
be chosen to be members of a complete set of functions. If the problem
formulation involves an
m
th order derivative, the trial function must be
u
→
u
as
m
→∞
(
m
−
1
)
times
continuously differentiable in the domain of concern.
One of the most familiar trial function approximations is the Fourier
1
series, where the
trial functions are composed of sine and cosine terms. Under appropriate conditions, many
functions can be conveniently and accurately represented by Fourier series in which the
trial functions form a complete set.
The trial function methods are often classified into
interior
and
boundary
procedures. In the
case of the interior method, the
N
ui
are chosen to satisfy the boundary conditions [Eq. (7.2)],
so that
u
satisfies the boundary conditions for all
u
i
. This is usually the easiest procedure
for most problems. Here, the
u
is to be determined such that the differential equations are
satisfied in some approximate sense. In the boundary method,
u
is chosen to satisfy the
governing differential equations [Eq. (7.1)], but not the boundary conditions. The problem
is reduced to that of selecting the parameters
u
i
such that the boundary conditions are
approximated. Another possibility is the combination of these methods for which the trial
solution
u
satisfies neither the boundary conditions nor the differential equations.
The methods of this chapter, which use trial functions of the form of Eq. (7.9), can be
categorized as belonging to either residual (or weighted residual) or variational methods.
1
Jean Baptiste Joseph Fourier (1768-1830) was a French physicist who, as the son of a tailor, was educated by the
Benedictines. His lack of good birth precluded his receiving a commission in the scientific corps of the French
army. He did, however, obtain a military lectureship in mathematics. He backed the revolution and accompanied
Napoleon on his 1798 Eastern expedition. He was given the post of governor of Lower Egypt, where he wrote
several papers on mathematics. After the 1801 French capitulation to Britain, Fourier returned to France and began
his experiments on the propagation of heat. His 1822 publication
Theorie analytique de la chaleur
, which dealt with
the flow of heat, contained the familiar contention that any continuous or discontinuous function of a variable
can be expanded in a trigonometric series of multiples of the variable.
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