Biomedical Engineering Reference
In-Depth Information
Define the solution array
∼
and derive the coefficient matrix
K
and the
array
f
∼
for the two element mesh depicted in the figure.
(g) Determine the solution array
∼
.
14.6 Using an isoparametric formulation, the shape functions are defined with
respect to a local coordinate system
−
1
≤
ξ
≤
1. Within an element the
unknown
u
h
is written as
n
u
h
=
N
i
(
ξ
)
u
i
.
i
=
1
(a) What are the above shape functions with respect to the local coordi-
nate system if a linear or quadratic interpolation is used?
(b) How is the derivative of the shape functions
dN
i
dx
obtained?
(c) Assume a quadratic shape function, and let
x
1
=
0,
x
2
=
1, and
x
3
=
3. Compute
dN
i
dx
for
i
=
1, 2, 3.
(d) Compute from array
∼
given by
3
d
∼
∼
=
dx
xdx
,
0
the first component using the same quadratic shape functions as
above.
14.7 Consider in the code
mlfem_nac
the directory
oneD
. The one-
dimensional finite element program
fem1d
solves the diffusion problem:
d
dx
c
du
dx
+
f
=
0,
on a domain
a
≤
x
≤
b
subject to given boundary conditions for a cer-
tain problem. The input data for this program are specified in the
m
-file
demo_fem1d
, along with the post-processing statements.
(a) Modify the
m
-file
demo_fem1d
such that five linear elements are
used to solve the above differential equation, using the boundary
conditions
u
=
0at
x
=
0,
and
