


The last time finiteelement meshing was in the headlines, developers were debating the advantages of tetrahedral (pyramid) versus hexahedral (brick) elements. Simulation experts today are talking about polyhedral elements, those with 12 to 14 faces and shaped like soccer balls. The unusual element comes from CDadapco Inc., Melville, N.Y., and London, England, and is said to work well on large CFD problems, but also shows promise for other simulations such as heat transfer/conduction and stresses in solids.
"And the fewer element surfaces in the grid, the shorter the solution time."
Tets and hexes were used because meshers could build grids of them and compute shape functions in the volumes. In the 1990s, researchers recognized it would be useful to extend meshing to more than six faces, the number on a brick or hex element. A polyhedral mesh is a good candidate for the next preferred element (or in the case of CFD, control volume) because it can be formed by combining several tets into a single polyhedron and is thus more flexible for filling the space with wellshaped elements.
Loworder elements, such as tets, are often deformed during meshing to look more like wedges or slivers. "Some of these point upstream, letting fluid flows hit their very oblique surfaces. The disparity between the small inflow area and large outflow areas leads to excessive numerical diffusion. Ideally, flow should be perpendicular to the element surfaces. Hence, tetrahedrons are the least ideal computational cell. And they are terribly inefficient at filling space. More tets are needed to fill a void than higher order elements with similar dimensions. Hexahedrons are an improvement, but they are difficult and expensive to generate automatically," says Nagy.
Polyhedral meshes take up to 30% more time to generate than tet meshes, but they fill space more efficiently, with fewer surfaces of a given size than other elements would require. "A 14sided polyhedral element, for example, always has a couple faces close to the flow normal. Hence, the solver finds accurate answers and sooner," says Nagy. "Tests show they produce the same answers as traditional meshes but in only 20% of the solution time and with half the memory required."
Socalled faceaddressing CFD solvers benefit from their ability to run polyhedral meshes, which contain fewer faces in a computational grid, for a given mesh resolution. Minimizing the number of faces boosts solver speed without sacrificing accuracy. Because each polyhedral cell has more faces than a tet or hex element, it also has more neighbors than traditional cell
types. The result, says Nagy, is that data propagates more quickly through polyhedral meshes, which leads to higher rates of convergence.
A Tale Of Tets And Polyhedrons  
Surface size (m) 
Tets 
Polys 
4.0 
95,050 
19,288 
2.00 
94,542 
19,187 
1.00 
191,873 
36,586 
0.50 
609,565 
112,471 
0.25 
3,351,276 
597,342 
Surface size refers to the height of a tet or diameter of a polyhedron. 
Of bubbles and polyhedrons
In 1887, Lord Kelvin found that a 14sided polyhedron (tetrakaidecahedron) would most efficiently pack bubbles into foam when all bubbles are of equal size. In 1994, physicists Denis Weaire and Robert Phelan found that a mixture of 12 and 14sided polyhedrons partition space 3% more efficiently than Kelvin's foam. The recent polyhedral meshes actually use an unrestricted number of faces, so they fill space in most efficiently. In fact, for a given resolution level, a mesh of polyhedral cells has fewer faces than a mesh of any other cell type. 
CDadapco Inc., cdadepco.com