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Published on *CD-adapco* (http://www.cd-adapco.com)

Stephen Ferguson

August 20, 2013, 4:20 am

In the words of the Cole Porter 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 neither do some CFD engineers.

Whether polyhedral meshes are the best for all types of CFD calculations still remains to be proved (although I 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 polyhedral meshes. 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 those CFD engineers (average brain size 1250g) who still religiously rely on tetrahedral meshing? 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 40 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 face addressing, 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.

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 0.3% 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 is 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 simulation 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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**Links:**

[1] http://www.cd-adapco.com/products/star-ccm%C2%AE

[2] http://www.cd-adapco.com/products/workflow

[3] http://www.cd-adapco.com/products/star-ccm%C2%AE/cad-mesh