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Dynamics >>25
Polyhedra:
Nature's answer to meshing
Stephen Ferguson, CD-adapco London
In
the words of a famous song: "Birds do it. Bees do it. Even
educated fleas do it." In reality, although bees are very good
at it, birds and fleas don’t do much polyhedral meshing –
but then up until recently, neither did most CFD engineers.
Whether
polyhedral meshes are the best for all types of CFD calculations,
it still remains to be proved although we think that the evidence
in their favor is rather compelling). Nature, however, has come
to her own conclusions and is full of examples of two and three-dimensional
tessellations that bear remarkable similarities to CD-adapco's latest
meshing technology. In contrast, there are relatively few examples
of naturally occurring hexahedral and tetrahedral mesh structures.
So
how is it that honeybees (average brain size 1g) manage to outmesh
the vast majority of CFD engineers (average brain size 1250g)? The
answer is obviously not that bees are more intelligent than engineers
(although there are a few notable exceptions).
Whereas CFD
and associated meshing technology has been around for just 30 years,
bees benefit from several billion years of evolution. From a purely
evolutionary viewpoint, the hexagonal structure of the honeycomb
is the endpoint of an exercise in energy optimization. The
walls of each honey cell are fashioned from wax and are manufactured
to a high tolerance (within 0.2% of their 100 micron thickness).
Creating this wax costs energy that could be better-used making
honey to rear the next generation of bees.
As
Charles Darwin himself wrote:
"With
respect to the formation of wax, it is known that bees are often
hard pressed to get sufficient nectar...it has been experimentally
proven that from twelve to fifteen pounds of dry sugar are consumed
by a hive of bees for the secretion of a pound of wax".
Darwin
also described the honeycomb as "a masterpiece of engineering"
that is "absolutely perfect in economizing labor and wax."
Biologists have long contended that the honeycomb was the ideal structure for containing the maximum amount of honey while containing the minimum amount of wax, however mathematical proof of this so-called "honeycomb conjecture" was a long time in coming. The conjecture, which has been a subject of mathematical curiosity since the third century AD, wasn't finally proved until June 1999, when Thomas C Hales of the University of Michigan, finally demonstrated conclusively that "a hexagonal grid represents the best way to divide a surface into regions of equal area with the least total perimeter”
As
good as honeybees might be at meshing in two dimensions, most practical
CFD work requires three-dimensional meshing. (This might offer a
degree of relief to any CFD engineers who are nervous about losing
their job to a swarm of inexpensive meshing bees.) In the same way
as bees benefit from minimizing the amount of wax used in producing
a certain volume of honey, face-addressing CFD solvers benefit from
minimizing the number of faces used in a computational mesh for
a given mesh resolution. [Using faceaddressing, the solver must
loop over all cell faces at every solution level – minimizing
the number of faces obviously has a huge payback in terms of solver
efficiency – see "The Advantage of Polyhedral meshing"
Dynamics 24.]
From this point of view, tetrahedra are the worst
type of computational cell. As the lowest order polyhedron they
fill space less efficiently than any other element. If bees worked
in three dimensions they wouldn't use tetrahedra, as the cost of
wax to honey would be too high. Although hexahedra are better from
this point of view, they too are far from the ideal.
The obvious question is therefore: Which type of mesh has the fewest number of faces per unit volume? Once again nature has the answer...
In
1887 Lord Kelvin became intrigued by the packing of bubbles in a
perfect foam – one in which all bubbles had equal volume.
He asked himself "How would bubbles of equal volume pack together,
to give the least possible amount of surface film between them?"
His answer was a 14-sided polyhedron that he painfully named
the "tetrakaidecahedron". This element was appealing because
it led to a regular symmetric partitioning of space – offering
an apparent improvement over nature, which uses a combination of
irregular polyhedrons in real soap foams. The tetrakaidecahedron
stood as the best way of partitioning space until 1994 when physicists
Denis Weaire and Robert Phelan rejected Kelvin's symmetrical partitioning
in favor of a more nature inspired solution. The so-called Weaire-Phelan
structure is a mixture of 12 and 14 sided polyhedra that partitions
space 3% more efficiently than Kelvin's foam.
What does this
have to do with CFD? Well, CD-adapco's polyhedral meshes typically
consist of cells of 12 and 14 faces (although the number of faces
is unrestricted). This means that they fill space in close to the
most efficient way possible. For a given resolution level, a mesh
consisting of CD-adapco's polyhedral cells has fewer faces than
a mesh of any other cell type.
Apart from the obvious benefits of economy, polyhedral meshes provide other advantages too. Because each polyhedral cell has more faces, it also has more neighbors than traditional cell types. A tetrahedral cell communicates with only four neighbor cells, and a hexahedral just six. In both cases this limits the influence of each cell to just a few neighbors. By contrast each polyhedral cell has on average 12 or 14 neighbors. The net result of this is that information propagates much more quickly through a polyhedral mesh, ultimately leading to an increased rate of convergence.
In the same way that a polyhedral cell "speaks" to more of its neighbors than other cell types, it also "listens" to information from more of them. Because each polyhedral cell receives information from more of its surroundings, the cell centered values calculated for the cell are more accurate than for other types.
The downside? We're not sure that there are any. While a flow fitted hexahedral meshes still offer some advantages, they are difficult and expensive to create (and if you know how to make your mesh truly flow fitted you probably don't need to run the calculation in the first place). Polyhedral meshes can be created at the click of a button and have advantages in efficiency and accuracy.
Sometimes nature knows best.
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