Biomedical Engineering Reference
In-Depth Information
by
1
r
u
r
=
0
,
u
rr
+
0
<
≤
r
<
L
,
(4.55)
u
r
(
)
=−
10
,
u
(
L
)
=
0
.
(4.56)
The exact solution is given by
u
(
r
)
=−
10
ln(
r
)
+
10
ln(
L
)
.
(4.57)
If we solve this equation for
=
0
.
01,
L
=
9 using a standard finite element
method with linear elements and with nodes at
r
=
0
,...,
9, the solution for
≤
r
<
1 is very unsatisfactory, as shown in Fig. 4.19. However, by using a simple
enrichment function
ψ
1
(
r
)
=
ln(
r
), and using this enrichment function on the
first two nodes (located at
r
=
0, 1), dramatically better results are achieved
(Fig. 4.19). Of course, refining the finite element mesh would also improve the
results, but this requires remeshing as the interface (in this example the left
boundary) moves. The X-FEM achieves this accuracy without remeshing.
The merits of coupling level sets to the extended finite element method
were first explored in [118], and subsequently its advantages further realized
in [53, 61, 82, 115, 117, 120]. The two methods make a natural pair of methods
where:
1. Level sets provide greater ease and simplification in the representation
of geometric interfaces.
2. The X-FEM, given the right enrichment functions, can accurately compute
solutions of elliptic equations which are often required for computing the
interface velocity.
3. Geometric computations required for evaluating the enrichment func-
tions (such as the normal or the distance to the interface) are readily
computed from the level set function [120].
4. The nodes to be enriched are easily identified using the signed distance
construction of the level set function [115, 117, 118, 120].
Compared to the other methods to follow, this algorithm is more complex,
but it is also much more general. Through the use of enrichment functions, this
method provides a much better solution near the interface, providing subgrid
resolution in that region without requiring additional mesh refinement. This is
