Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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<strong>Adaptivity</strong> <strong>with</strong> <strong>moving</strong> <strong>grids</strong> 13<br />
1<br />
Ω C<br />
1<br />
Ω P<br />
0.9<br />
0.9<br />
0.8<br />
0.8<br />
0.7<br />
0.7<br />
η<br />
0.6<br />
0.5<br />
y<br />
0.6<br />
0.5<br />
0.4<br />
0.4<br />
0.3<br />
0.3<br />
0.2<br />
0.2<br />
0.1<br />
0.1<br />
0<br />
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1<br />
ξ<br />
0<br />
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1<br />
x<br />
Figure 2.1. A typical map (x(ξ,η),y(ξ,η)) from a<br />
computational domain Ω C to a physical domain Ω P .<br />
2006), the boundary map is obtained automatically as part of the algorithm.<br />
This is an attractive feature from the perspective of algorithmic complexity,<br />
although it does lead to a reduction of control of the boundary points.<br />
The great merit of this approach is that it transforms the problem of<br />
finding (and describing) a mesh in Ω P (which in the case of h-adaptive<br />
methods can require subtle data structures, including hierarchical trees)<br />
into the much simpler problem of describing the function F. Much of this<br />
article is devoted to deriving suitable equations for F and seeking effective<br />
solution strategies for them. It goes <strong>with</strong>out saying that it should not be<br />
more difficult to find F than to solve the underlying PDE, and indeed that<br />
many such functions F may give appropriate meshes on which to solve<br />
the PDE. The properties of the mesh τ P then follow immediately from the<br />
structure of the map F. This simple observation is a key to the success<br />
of r-adaptive methods, as it allows the use of powerful mathematical tools<br />
to describe, construct and control the mesh behaviour. These include the<br />
application of methods from differential geometry (especially the theory<br />
of optimal transport) to describe the static structure of τ P , and methods<br />
from the theory of dynamical systems to describe its evolution. The latter<br />
is especially appropriate when coupled to the partial differential equations<br />
which are often solved to find F. The unity that r-adaptive meshes give<br />
for both solving the underlying PDE, and finding the mesh, is a significant<br />
advantage of r-adaptivity over other adaptive approaches.