Biomedical Engineering Reference
In-Depth Information
function
u
(
x
):
R
d
d
assumes the form
→ R
u
h
(
x
)
=
d
)
,
φ
I
(
x
)
u
I
,
(
u
I
∈ R
(4.53)
I
n
I
∈
N
where the functions
φ
I
(
x
) are the finite element basis functions and
u
I
are the
weights.
The extended finite element method uses enrichment functions, extra basis
functions which are sensitive to prescribed boundaries, to capture the boundary
conditions and improve the solution in the neighborhood of regions which would
otherwise require greater spatial resolution. Consider again a point
x
that lies
inside a finite element
e
. The enriched approximation for the function
u
(
x
)
becomes
u
h
(
x
)
=
φ
I
(
x
)
u
I
φ
J
(
x
)
ψ
(
x
)
a
J
(4.54)
+
,
I
n
I
∈
N
J
n
J
∈
N
g
classical
enriched
where the nodal set
N
g
consists of nodes which are on elements cut by the
boundary, for example, see Fig. 4.18. In general, the choice of the enrichment
function
ψ
(
x
) that appears in Eq. 4.54 depends on the geometry, the boundary
condition, and the elliptic equation being solved.
To illustrate the effectiveness of this approach, consider the following simple
example. Suppose we wish to solve the radial heat equation on an annulus given
Figure 4.18:
Example of choosing enriched nodes. Enriched nodes are indi-
cated by gray dots.
