[Jonas Holdeman]

I am trying to gather some background information on use of triangular prismoidal (TP) finite elements in general, and Hermite FEs in particular. Where and how are they being used, in CFD or elsewhere? I would appreciate it if you might help with your experience, comments, and/or references. References available on the internet would be the most useful as I don't have ready access to a library.

Some obvious uses of TP elements would be in cylindrical domains, or for tiling around a boundary in 3D for transition to tetrahedral elements internally. But how else might they be used effectively?

Actually, my application is for 2D periodic incompressible flows. I use modified divergence-free velocity elements which are the curl of a scalar Hermite stream function element. The degrees-of-freedom are the stream function, two (divergence-free) velocities in space dimensions, and one DOF in time, with periodic BCs in time. I am now using hexahedral elements this way, and anticipating using TP elements for comparison. The special domain would be partitioned into triangles, with the axial freedom being time.

Again, your help would be greatly appreciated.

[mprinkey]

Any transition from hexahedral cells to tetrahedral cells will involve a layer of pyramids. That is a topological necessity--converting quad faces to tri faces. Honestly, that is the only application I can think of. Wedges are much more commonly used for building boundary or interface layers by extruding triangulated surface meshes.

Pyramids are just not topologically useful for much besides glue--they can't be stacked on their own in any meaningful way...Two pyramids quad-to-quad gives you an icosahedron with all tri faces. A cluster of six pyramids all tri-to-tri nets you a hex--all quad faces. Using those templates, you would be able to space fill with clusters of pyramids but probably only by accepting worse aspect ratios than using tets or hexs directly. I guess too they could be used in a conformal way to refine meshes, but I think that there are better ways using tets or hexs directly.

[Jonas Holdeman]

Thank you Michael. I think your wedges are the same as my triangular prisms. Your term sounds more familiar, so I will use it in the future. Also what I referred to as tiling, you better described as extruding from a triangular surface mesh.

So my question becomes, does anyone know of anyone using Hermite wedge elements, and in what context? Now, if I could just change the title of this thread ...

[mprinkey]

Wow, I apologize. I misread that entirely. I was stuck on pyramids.

Yes, "wedges" are the ANSYS/Fluent speak I am most familiar with. As I said, wedges are great for extruded boundary layers or anywhere you need refinement at a natural interface. Generally, they are a nice way to take a surface/interface triangularization and generate surface-normal resolution.

They are also useful in rounding out harsh corners in primarily hexahedral meshes. Wedges are used to fill-in/round-off edges while tets fill-in/round-off corners. I may think of more.

Sorry again for the confusion.

[FMDenaro]

Originally Posted by Jonas Holdeman

I am trying to gather some background information on use of triangular prismoidal (TP) finite elements in general, and Hermite FEs in particular. Where and how are they being used, in CFD or elsewhere? I would appreciate it if you might help with your experience, comments, and/or references. References available on the internet would be the most useful as I don't have ready access to a library.

Some obvious uses of TP elements would be in cylindrical domains, or for tiling around a boundary in 3D for transition to tetrahedral elements internally. But how else might they be used effectively?

Actually, my application is for 2D periodic incompressible flows. I use modified divergence-free velocity elements which are the curl of a scalar Hermite stream function element. The degrees-of-freedom are the stream function, two (divergence-free) velocities in space dimensions, and one DOF in time, with periodic BCs in time. I am now using hexahedral elements this way, and anticipating using TP elements for comparison. The special domain would be partitioned into triangles, with the axial freedom being time.

Again, your help would be greatly appreciated.

Hello,
I am not expert in FE but I as I know they are best suited wherever elliptic problems are present...
from a math point of view: http://file.scirp.org/Html/6-7401945_41077.htm

in CFD, I think you could find useful the book of Chung:

https://books.google.it/books?id=Cq6...ermite&f=false

[Jonas Holdeman]

Thank you for the references, Filippo.

I am really interested now in the 3D triangular prism or wedge elements. The first reference deals with a 2D triangular element, and to be frank, the result is rather restricted and there are better finite elements for that problem. According to the Table of Contents of the Chung book, the TP/wedge elements are referenced in section 9.4.2 on page 302, but that page is missing in your link. Since only one page of a 1000+ page book is allotted to TP/wedge elements, the treatment must be rather terse. But that fact may be significant.

I have a legitimate 24 DOF, quadratic-complete Hermite TP/wedge element that I worked out mostly by inspection from a (2D) Hermite triangular and a (3D) Hermite hexahedral element, but it is not as efficient as I would like. It would take several days or more hard work for me to derive it right. I also have a 8-node, 32 DOF, complete cubic element that has similar limitations. I am working on some (incompressible fluid) problems with moving mesh & boundary - piston-driven and peristaltic flow - and would like to use the TP/wedge element for comparison. That is why I am searching for background info on the latter type elements.

Thanks again.

[Rami]

Hi Jonas,

Don't Hermitian elements necessarily imply C1 continuity (e.g., shells in solid FEM)? If so, I don't see its justification in CFD, where C0 continuity holds.

Regards,
Rami

[Jonas Holdeman]

Originally Posted by Rami

Hi Jonas,

Don't Hermitian elements necessarily imply C1 continuity (e.g., shells in solid FEM)? If so, I don't see its justification in CFD, where C0 continuity holds.

Regards,
Rami

The Hermite elements with first derivatives for the stream function P may be C0 continuous across adjacent elements. Taking the curl (differentiating) to get vector elements maintains continuity in the normal direction, but not tangential continuity (u=dP/dy, v=-dP/dx, with similar results in 3D). The Hermite elements with all first and second derivatives may be C1 continuous. Taking the curl to get vector elements leaves C1 continuity in the normal direction, but C0 continuity in the tangential direction across boundaries, but are still C1 at the nodes.

But just being Hermite (including derivatives) does not imply continuity. NOTE: I am calling any element that includes derivatives Hermite, as distinguished from Lagrange elements that do not include derivatives.