Biomedical Engineering Reference
In-Depth Information
It is clear that in Eq. (
14.60
) the unknowns are
u
2
,
u
3
,
u
4
on the left-hand side
of the equation, and
p
u
on the right-hand side of the equation. So both columns
can be divided in a known part and an unknown part, depending on the essential
and natural boundary conditions that have been prescribed. The next section will
outline how this equation is partitioned to facilitate the solution process.
14.6
Solution of the discrete set of equations
Let Eq. (
14.60
) be written as
K
∼
=
f
∼
(14.61)
where
f
ext
.
The unknowns can be partitioned into two groups. First, some of the compo-
nents of the column
∼
will be prescribed. This subset of
∼
∼
=
f
int
+
f
∼
∼
is labeled
∼
p
. The
remaining components of
∼
are the actual unknowns, labeled
∼
u
. In a similar
manner
K
and
f
∼
can be partitioned. Consequently, Eq. (
14.60
) can be rewritten as
K
uu
K
up
K
pu
K
pp
∼
u
∼
p
f
∼
.
u
=
(14.62)
f
∼
p
It is emphasized that the right-hand side partition
f
∼
u
associated with the unknowns
∼
u
will be known, and reversely,
f
∼
p
will be unknown as
∼
p
is known. Since
∼
p
is
known, the actual unknowns
∼
u
can be solved from
K
uu
∼
u
=
f
∼
u
−
K
up
∼
p
.
(14.63)
Notice that at the part of the boundary where
u
is prescribed, the associated
external load
f
p
is unknown.
The components of
f
∼
p
can be obtained by simple multiplication, after having
solved
∼
u
from Eq. (
14.63
):
f
∼
p
=
K
pu
∼
u
+
K
pp
∼
p
.
(14.64)
14.7
Isoparametric elements and numerical integration
In Section
14.4
the concept of shape functions has been introduced. Within each
element
u
h
has been written as
T
(
x
)
∼
e
.
u
h
|
e
=
∼
(14.65)
