Civil Engineering Reference
In-Depth Information
prescribed boundary conditions and an integral formulation to minimize error, in
an average sense, over the problem domain. The general concept is described
here in terms of the one-dimensional case but, as is shown in later chapters,
extension to two and three dimensions is relatively straightforward. Given a
differential equation of the general form
D
[
y
(
x
),
x
]
=
0
a
<
x
<
b
(5.1)
subject to homogeneous boundary conditions
y
(
a
)
0
(5.2)
the method of weighted residuals seeks an approximate solution in the form
=
y
(
b
)
=
n
y
*(
x
)
=
c
i
N
i
(
x
)
(5.3)
i
=
1
where
y*
is the approximate solution expressed as the product of
c
i
unknown,
constant parameters to be determined and
N
i
(
x
)
trial functions. The major
requirement placed on the trial functions is that they be
admissible functions;
that is, the trial functions are continuous over the domain of interest and satisfy
the specified boundary conditions exactly. In addition, the trial functions should
be selected to satisfy the “physics” of the problem in a general sense. Given these
somewhat lax conditions, it is highly unlikely that the solution represented by
Equation 5.3 is exact. Instead, on substitution of the assumed solution into the
differential Equation 5.1, a residual error (hereafter simply called
residual
)
results such that
0
(5.4)
where
R
(
x
) is the residual. Note that the residual is also a function of the
unknown parameters
c
i
. The method of weighted residuals requires that the
unknown parameters
c
i
be evaluated such that
b
R
(
x
)
=
D
[
y
*(
x
),
x
]
=
w
i
(
x
)
R
(
x
)d
x
=
0
i
=
1,
n
(5.5)
a
where
w
i
(
x
)
represents
n
arbitrary weighting functions. We observe that, on
integration, Equation 5.5 results in
n
algebraic equations, which can be solved for
the
n
values of
c
i
. Equation 5.5 expresses that the sum (integral) of the weighted
residual error over the domain of the problem is zero. Owing to the requirements
placed on the trial functions, the solution is exact at the end points (the boundary
conditions must be satisfied) but, in general, at any interior point the residual
error is nonzero. As is subsequently discussed, the MWR may capture the exact
solution under certain conditions, but this occurrence is the exception rather than
the rule.
Several variations of MWR exist and the techniques vary primarily in how
the weighting factors are determined or selected. The most common techniques
are point collocation, subdomain collocation, least squares, and Galerkin's
