From: a722756@roper.mc.ti.com (W. Donald Rolph)
Subject: Re: Mesh generation schemes (was Re: FEA help!)
Originator: a722756@roper.mc.ti.com
Organization: Texas Instruments / Attleboro Mass / USA
Date: Thu, 17 Dec 1992 18:56:13 GMT


In article <1992Dec16.233849.5900@utagraph.uta.edu>, muthu@mecad.uta.edu (S.N. Muthukrishnan) writes:
|> In article <1992Dec16.164320@roper.mc.ti.com> a722756@roper.mc.ti.com (W.  
|> Donald Rolph) writes:
|> > 
|> > In article <1992Dec15.225304.11571@porthos.cc.bellcore.com>,  
|> bhatia@bigcat.bellcore.com (Raman Bhatia) writes:
|> > |> In article <1992Dec14.181725@roper.mc.ti.com>, a722756@roper.mc.ti.com (W.  
|> Donald Rolph) writes:
|> > |> |> 
|> > |> |> This is by no means a trivial question.  You must first ask the class  
|> of FEA to
|> > |> 
|> > |> Raman Bhatia (bhatia@bigcat.bellcore.com)
|> > 
|> > I have an interesting proof that show that the triangularization schemes all
|> > violate controllability theory at any given point in the model.  The effect  
|> is
|> > that in 3D, with typcial dimensions in orthoganoal directions which have
|> > significantly different scales (1.4 in thick by 5 inches wide by 20 inches  
|> long),
|> > the triangularizations schemes produce highly unsatisfactory meshes for many
|> > classes of problems, particularly nonlinear problems.  It is interesting to  
|> note
|> > the CFd codes are gradually adopting Laplacian mesh generation schemes,  
|> schemes
|> > which were popular as early as 1960 for fea.  It should be noted that CFD can  
|> be
|> > very suseptible to poor meshes, and perhaps there is a message here!  
|> > 
|> > I have simplistic proof which suggests that Laplacian mapping produces the
|> > "optimal mesh" fgor certain classes of problems, based on the givven mesh  
|> density
|> > provided as input.  I also have s ample program which generates hexahedral  
|> meshes
|> > off an arbitray boundary rep trimmed nurb solid model, demonstrating that the
|> > technogy is feasible, and the input is not overly burdensome.
|> > -- 
|> > 
|> > Regards.
|> >  
|> > Don Rolph  a722756@pan.mc.ti.com WD3 MS10-13 (508)-699-1263
|> 
|> Yes. That is true. However, it is possible to control the "aspect ratio"
|> in meshes, be it 2-D or 3-D. I assume that in 2-D it is triangular elements
|> and in 3-D it is tetrahedral elements. Laplacian mapping is one way to improve
|> the aspect ratio. There is also a very good paper in mesh smoothing. I shall
|> mail it or post it if there is any interest. I don't remember it off my head.
|> 
|> -Muthu
|> --
|> ---------------------------------------------------------------------
|> S. N. Muthukrishnan                | muthu@mecad.uta.edu |
|> Dept.of Mech. Engg., UT, Arlington | muthu@fem.uta.edu   |(NeXT mail)
|> Arlington, TX  76019               | muthu@newton.uta.edu|
|> _____________________________________________________________________
|> "Perception is a reality. Reality is myth"
|> ---------------------------------------------------------------------
 
A critical point is that one really needs hexahedral elements to get well
performing meshes in 3D.  The automeshers proposed so far admit this limitation. 
Further, it is worth noting that the mesh should map to the solution,  that is
the best results will occur when the principal directions of the solution
variation occur along element edges (there is an interesing picture in one of
Zienkiewiczs books which show the displacement field behavior for the clasic
serendipity quadratic element:  the field diagonal to the element is not pretty!)

When these observations are combined with the controllability arguement, I find
the case for triangularization schemes less than compelling.

Would like to see your smoothing article though!!
-- 

Regards.
 
Don Rolph  a722756@pan.mc.ti.com WD3 MS10-13 (508)-699-1263
Newsgroups: sci.math.num-analysis