Biomedical Engineering Reference
In-Depth Information
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Figure 4.16:
Comparison of the old (left) and new (right) velocity extension
methods.
the interface slows down when it approaches the left side, while with the new
method the interface wraps around and merges. The two solutions are shown
side by side in Fig. 4.16. The characteristics for this example, represented by the
lines of constant F , are shown in Fig. 4.17, illustrating the analogous solution as
computed in the theorem. Note how the lines of constant F are orthogonal to
the lines of constant φ , as a result of solving Eq. 4.41.
4.3.3 Coupling to Elliptic Solvers
Very often, the speed of the interface is determined by solving an associated
elliptic equation, e.g. the pressure equation for incompressible fluid flow. This
leads to an elliptic equation which must be solved on an irregularly shaped
domain or where there is an internal boundary with jump conditions across the
boundary. There are several strategies to handle this problem. When using finite
elements to solve this elliptic equation, a mesh is dynamically generated so that
it conforms to this irregular boundary. When using finite differences, special
delta functions can be added at nodes near the interface to enforce the jump
conditions, see e.g. [88].
In the context of the level set method, there are three strategies for set-
ting up and solving the associated elliptic equation. They vary in generality,
 
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