Adaptivity with moving grids

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