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Selina Tracy March 22, 2001 17:16

Egyption Pyramid with All HEXA
Is there anyone who can make Egyption pyramid with all hexa meshes. I've been trying for a couple of weeks but only to fail.



John C. Chien March 22, 2001 18:12

Re: Egyption Pyramid with All HEXA
(1). I thought that the big stone blocks used in Egyptian pyramid were six-sided. (2). I think everybody knows that they were far more advanced several thousand years ago.

Selina Tracy March 22, 2001 20:21

Re: Egyption Pyramid with All HEXA
Do you really think all of the bricks are six sided. What about the very point of apex. Thanks.


John C. Chien March 23, 2001 00:15

Re: Egyption Pyramid with All HEXA
(1). A point can be a small square. Even a needle point is actually rounded.(2). There is no way one can compute flow around a point. This is exactly the reason why the computational geometry is not the CAD geometry. (3). In CFD, the geometry and the mesh are handled together, and both must be modelled.

Selina Tracy March 23, 2001 02:55

Re: Egyption Pyramid with All HEXA
Topology for Selina and CFD for John.


John C. Chien March 23, 2001 03:56

Re: Egyption Pyramid with All HEXA
(1). A finite length cylinder with constant diameter is a round cylinder. (2). A finite length cylinder with zero diameter at the top is a cylindrical cone. (3). A finite length square with constant cross-sectional shape is a square cylinder. (4). A finite length square with a zero cross-sectional area at the top is a pyramid. (5). They all have the same topology. The cross-sectional shape is self-similar.

Krylov K Gorky March 23, 2001 04:03

ICEM etc: Egyption Pyramid with All HEXA
i've just send an email to icem etc. to see if they can do it. but i doubt it. it's not a sort of question that a company can give an answer. it's too academic if you stick to math. let's see. googluck to all. -Kryl

Frank March 23, 2001 05:19

Re: ICEM etc: Egyption Pyramid with All HEXA
Very funny to read your discussion.

What about simply deviding a hexa into 6 pyramids. It's base consists of the hexa faces and the endpoint of the centroid. The faces are described via bi-linear shape functions (forget about splitting a face, that has 4 vertices in to 2 triangles -> that is the stupid way to do it). With those pyramids you can build up every topology you like. Not only hexa, but also degenerate a quafrilateral face into a triangle.

Taking a structured flow solver and degenarating a hexa into a pyramid is probably not a good thing to do, since structured solvers are very picky, due to utilized metrix terms (not computing in physical space looses rubustness)


Joern Beilke March 23, 2001 07:03

Re: Egyption Pyramid with All HEXA
Just divide the pyramide into 2 tets. Now each tet can be divided into 4 hex-blocks.

I just created a mesh using this topology. If anybody is interested I can put the stuff somwhere.

Joern Beilke March 23, 2001 07:40

Re: Egyption Pyramid with All HEXA
Here are the files:

and the mesh in Grid3d and Nastran format:

Have fun

John Chawner March 23, 2001 09:50

Re: Egyption Pyramid with All HEXA
There are at least three topologies that come to mind immediately.

1. Split the bottom edge of each triangular face. Then each triangle is now four sided and you insert a quad mesh.

2. Insert a node into the middle of each tri face and then bisect each edge of the tri face. Now the tri faces are subdivided into three quad regions.

3. Use a singularity at the vertex.

-- John Chawner - - Gridgen: Reliable CFD Meshing

John C. Chien March 23, 2001 12:01

Re: Egyption Pyramid with All HEXA
(1). Very nice. (2). It is a multi-block, structured hex-mesh.

Selina Tracy March 23, 2001 13:37

Joern: Egyption Pyramid with All HEXA

But I have to say I did not defined my problem well.

What I meant was generating pyramid with all hexa while keeping only ONE FACE BASE.

Sounds TOUGH!

Could you try one more time if you can do it easily?


Selina Tracy March 23, 2001 13:40

Re: Egyption Pyramid with All HEXA
Joern did a wonderful job. But I had to redefine my problem. Let's Make the pyramid with all hexa while keeping ONE FACE BASE.



John C. Chien March 23, 2001 14:21

Re: Joern: Egyption Pyramid with All HEXA
(1). I had already given you the answer, that is, you create a square cylinder,create a single block hex mesh in it and then transform the mesh such that the tip area goes to zero. Then you will have one single block hex mesh. (2). That is the standard answer.

Robert Schneiders March 26, 2001 03:32

Re: Egyption Pyramid with All HEXA



Regards, Robert

Robert Schneiders MAGMA Giessereitechnologie GmbH D-52072 Aachen Kackertstr. 11 Germany Tel.: +49-241-88901-13 email: www:

Peter Bailey March 29, 2001 05:41

Re: Selina: Egyption Pyramid with All HEXA





Dear Selina and Robert,

I have solutions for All-hex meshes of both of your Egyptian Pyramid Patterns,WITHOUT degenerated elements, AND with interior angles between faces which are less than 180 degrees, AND with interior angles between edges which are less than 180 degrees.

The element shapes for the solution of Selina's problem,are,in fact,relatively good.

Both these Egyptian pyramid problems are very tough indeed.

For Selina's problem,with ONE quadrilateral face at the base,as in her 23rd March 2001 amendment to the problem,the solution involves 43 hexahedra.

Here is the solution,at least for the really difficult bits,for Selina's Egyptian Pyramid Problem,with ONE quadrilateral at the base...

First use Scott Mitchell's excellent 'Geode' template. You can get a copy of Scott Michell's paper on his Geode template from his Sandia labs based web-site,probably at ...

BEWARE: Scott Mitchell/Sandia have PATENTED the GEODE Template,at least in the US,although it is probably not patented,nor,possibly,patentable in the UK or various other countries.

Now place a pyramid on top of the Geode template, then split it in to 2 tetrahedra,to match with the the base quadrilateral face split along a diagonal to match the top diagonal of the Geode Template. Then split these upper 2 tetrahedra into 4 hexes each.This upper construction of 8 hexahedra is shown,in fact,in Joern Beilke's earlier communications to the CFD forum.

Now we imagine a cube placed at the bottom of the Geode Template,and split this cube in to 9 hexahedra,in such a way that it provides a transition from the 4 quadrilateral faces at the bottom of the Geode template,to the single quadrilateral face required at the bottom of the Egyptian pyramid by Selina.

Time constrains me from describing this bottom 9 hexahedra transition pattern in detail,which I have definitely found,but it should not be too difficult for someone with a reasonable amount of experience with hexahedral patterns to derive for his/her self.

Now we carefully squeeze the whole pattern from, essentially,a pyramid on top of 2 cubes,in to just one Egyptian pyramid,taking care to move the internal nodes accordingly to preserve the best shapes.

In fact,I have only imagined this pattern so far, and not had time to check it out in detail.The last 'squeezing stage' looks a bit dangerous as regards the effects on the Geode part of the pattern,but I think it looks as if the hexes in the geode pattern should still be okay,depending on how steeply the pyramid sides slope.We can also squeeze the geode part down towards the bottom of the pyramid in order to make its hex shapes better.

I do have an alternative to the Geode pattern for use in Selina's problem,but my Geode alternative is radically different and involves many more hexahedra - around 300 more at least - although it may have better shaped hexahedra.

Now to Robert Schneider's problem.This is probably the tougher problem,but I have a solution.It involves 298 Hexes in my original pattern,and 470 Hexes in a later pattern that has better shapes. The hexahedra are not degenerated,but the internal angles,between faces and/or edges,for some of the elements,go down to as little as around 10 to 15 degrees. Such small angles may be unnacceptable for some analysis packages,and situations,unless,perhaps such a pyramid template can be kept away from crucial areas of an overall mesh,and,somehow,used sparingly.

I also have an alternative pattern of all hexahedra which can probably be of more use than the pyramid pattern in producing all hex,or even hex-dominant, meshes,automatically,in conjunction with splitting up any pyramid shapes in simpler ways in to all hexahedra. My alternative pattern is called the X-PLUS,or X+ pattern,and it could also be used as an alternative to the Scott Mitchell's GEODE pattern.

The X+ pattern uses more hexahedra - 312,which is more than the GEODE pattern,but the X+ pattern is more symmetric,which means that it can probably fit in to more situations more easily for use in automatic hex mesh generators,e.g.using methods like Schneiders Octree based Hex mesh generator. The X+ pattern has much better shaped hexahedra than my 'Schneider's Pyramid' pattern.

Time is just one reason why I cannot provide more details about my patterns now.

However,Selina,and anyone else interested,should be able to get the details of Scott Mitchell's GEODE template from the Internet,and work out how to fit in the other bits that I have described,to solve Selina's Pyramid Problem. Perhaps Scott Mitchell at Sandia would also be interested ?

I would be very interested to know Selina's application - why does she need only one quadrilateral at the base ? Is it for some automatic mesh generator algorithm ?

For practical CFD situations,PHOENICS from CHAM Ltd.,where I currently work,can model and analyse such pyramid situations using BFC ( Body Fitted Coordinate),MultiBlock Grid Meshes.

Any queries for further information are most welcome.


Peter Bailey.

Selina Tracy March 30, 2001 04:15

Re: Selina: Egyption Pyramid with All HEXA
Your answer is one of the best that I have got for this geometry. Yes, I have already read the papers you recommended. But still the clouds are hanging over my head. I am working on an alogrithm which should come from "the simplest recursive formula" so that it can be used practical purpose. Inner topology that has all the features of geometry. Sounds too optimistic.


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