Biomedical Engineering Reference
In-Depth Information
The approximate solution of Eq. (
14.1
) is found by transforming the differential
equation into a discrete set of ordinary equations:
c
du
dx
d
dx
+
f
=
0
−→
K
∼
=
f
∼
.
(14.4)
The array
∼
contains
approximations
of the continuous solution
u
of the differen-
tial equation at a
finite
number of locations on the
x
-axis. Increasing the number
of points defining
∼
should lead to an increased accuracy of the approximation of
u
. A particularly attractive feature of the Finite Element Method is that the spatial
distribution of these points does not need to be equidistant and can be chosen such
that accurate solutions can be obtained with a limited number of points, even on
complicated geometries (in the multi-dimensional case) or problems with large
gradients in the solution.
The finite element method proceeds along three well-defined steps.
(i) Transformation of the original differential equation into an integral equation by
means of the principle of
weighted residuals
.
(ii)
Discretization
of the solution
u
by interpolation. If an approximation of the solu-
tion
u
is known at a finite number of points (nodes) an approximation field may be
constructed by interpolation between these point (nodal) values.
(iii) Using the discretization the integral equation is transformed into a
linear set of
equations
from which the nodal values
∼
can be solved.
14.3
Method of weighted residuals and weak form of the model problem
First of all the differential equation is transformed into an integral equation by
means of the
weighted residuals
method. Suppose that a given function
g
(
x
)
=
0
on a certain domain
a
≤
x
≤
b
, then this formulation is equivalent to requiring
b
w
(
x
)
g
(
x
)
dx
=
0
for all
w
,
(14.5)
a
and to emphasize this important equivalence:
b
g
(
x
)
=
0 n
a
≤
x
≤
b
⇔
w
(
x
)
g
(
x
)
dx
=
0
for all
w
.
(14.6)
a
The function
w
(
x
) is called the
weighting function
, and is assumed to be a con-
tinuous function on the integration domain. The equivalence originates from the
requirement that the integral equation must hold for
all
possible weighting func-
tions
w
. It therefore should also hold for
w
=
g
(
x
). For this particular choice of
w
, the integral expression yields
