Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
<strong>Adaptivity</strong> <strong>with</strong> <strong>moving</strong> <strong>grids</strong> 21<br />
choice of adaptivity strategy, gross mesh distortion or problems when the<br />
<strong>moving</strong> mesh interacts <strong>with</strong> a fixed boundary. In a purely Lagrangian setting<br />
a <strong>moving</strong> mesh used to calculate (for example) a fluid flow might seek to<br />
have v equal to the local velocity of the fluid particles. In practice, as we will<br />
demonstrate, such a procedure can lead to mesh tangling in the presence<br />
of flows <strong>with</strong> high vorticity, and mesh racing when the fluid particles leave<br />
the boundary and the mesh labelling has to be reassigned. In practice (and<br />
in a manner to be made precise presently) it is often optimal for the mesh<br />
points to move in a similar manner to the particles, but not to follow their<br />
motions too precisely.<br />
The connectivity of a mesh reflects how adjacent nodes are connected<br />
together. In an h-adaptive mesh connectivity can be a significant issue,<br />
and changes every time a local mesh refinement step is implemented. This<br />
causes additional overheads in setting up the equations of any discretization<br />
on this mesh, as the connectivity matrix needs to be constantly updated.<br />
In contrast, in an r-adaptive method, the connectivity of the <strong>moving</strong> mesh<br />
is usually determined by the connectivity of the underlying computational<br />
mesh, which usually does not change during the calculation, and we can<br />
presume to be relatively simple. A significant benefit of this approach is<br />
that various mesh-smoothing methods can use this constant connectivity<br />
and can (for example) exploit fast spectral methods which take advantage<br />
of the constant mesh connectivity in the computational domain. As an<br />
example, if the functions (x(ξ,η),y(ξ,η)) determine a particular mesh, then<br />
it is possible to construct a smoother mesh from this. One example is<br />
given by<br />
(ˆx, ŷ) = ( ) −1(x,<br />
I − γ∆ ξ y), (2.14)<br />
where ∆ ξ is the Laplacian operator applied in the computational domain and<br />
γ is chosen appropriately. An important reason for constructing a smoother<br />
mesh is to avoid significant variation in mesh size between adjacent elements.<br />
We presently consider the effect of this on the solution error. This procedure<br />
was introduced by Huang and Russell (1997a). The Laplacian operator can<br />
be inverted very rapidly on a simply connected uniform rectangular mesh by<br />
using a fast spectral method, for example the fast cosine transform. This<br />
smoothing procedure also damps out the creation of certain chess board<br />
modes that can lead to a deterioration of the mesh quality.<br />
2.4. Mesh topology<br />
The discussion so far has been restricted to the use of <strong>moving</strong> meshes mapping<br />
one convex region, indeed logically rectangular regions, to logically<br />
rectangular regions. There is no real problem mapping a logically rectangular<br />
region Ω C to another convex region. For example, the article by<br />
Calhoun et al. (2008) describes in detail how a logically rectangular mesh