Anisotropic Mesh Coarsening
For anisotropic problems -- including notably external viscous flow
problems -- there is often a pseudo-structured part of the mesh to
accurately resolve anisotropic physics. The pseudo-structured part
of the mesh typically contains quadrilaterals, prisms, or hexahedra
that have been divided into triangles or tetrahedra. Pseudo-structured
meshes are useful in treatment of anisotropic physics, and these pseudo-structured
regions should be preserved in coarse meshes to the extent feasible.
These pseudo-structured mesh regions can be coarsened isotropically,
reducing the number of vertices by a factor of in these regions
by removing alternate planes of points in each direction. However,
some work [5,12] suggests that multigrid
methods are much more efficient if coarsening is done anisotropically,
reducing the cell aspect ratio near the wall and the associated numerical
stiffness. We allow several variations on anisotropic coarsening to
accommodate different scenarios for surface and interior mesh pseudo-structure,
as discussed in Section 6.3.1.
- Three-dimensional meshes may have a locally anisotropic,
pseudo-structured surface mesh, as shown in Figure 6.3.
Roughly speaking, in this case, all fold vertices are retained, and
alternate vertices are retained along closely-spaced lines leaving
the fold. This process is described in detail in Section 6.3.1.
- Both two- and three-dimensional meshes may have
sections of locally anisotropic, pseudo-structured interior
mesh, similar in two dimensions to the example in Figure 6.3.
Pseudo-structured interior mesh fragments are coarsened in much the
same way as pseudo-structured surface mesh fragments.
Selection of Points in Pseudo-structured Anisotropic Meshes
Pseudo-structured surface mesh on a wedge.
In selecting points with pseudo-structured anisotropic mesh fragments
(PSAMF's) to retain in coarse meshes, we consider three important
and distinct cases:
These rules are simple to describe and simple for a person to follow,
but some care is required so that a computer can reliably identify
a pseudo-structured anisotropic mesh fragment and coarsen it properly.
For details, see .
- The fully-anisotropic case (3D). Some three
dimensional meshes have anisotropic layers built out from anisotropic
surface triangulations. A typical example of this is found in computational
aerodynamics, where cells near aircraft wings for viscous simulations
are much larger in the spanwise direction than in the streamwise direction,
and much larger in the streamwise direction than normal to the wing
surface. To reach unit aspect ratio in both directions during coarsening,
we initially keep every point along the trailing edge in the spanwise
direction, alternate lines on the wing surface, and every fourth layer
in the interior of the mesh. This reduces the aspect ratio in both
directions, and reduces the number of vertices by a factor of 8 in
these mesh fragments. As the spanwise-streamwise aspect ratio approaches
one (i.e., as the surface mesh becomes isotropic), such a mesh
fragment would revert to Case 2.
- The semi-anisotropic case (3D). Some three
dimensional meshes have anisotropic layers built out from isotropic
surface triangulations. In this case, we choose to coarsen the surface
mesh as we would a two-dimensional isotropic mesh. In the interior,
we choose to emphasize reduced cell aspect ratio near the boundary
by selecting every fourth vertex along marching lines beginning at
the surface vertices that will be retained. This approach reduces
the aspect ratio of anisotropic cells near the boundary by roughly
a factor of two at each coarsening and reduces the number of vertices
in such mesh fragments by a factor of 16.
- In two dimensions, directional anisotropic coarsening
should select every fourth vertex along marching lines extending from
each point on the boundary into the domain.