[Gmsh] Two questions:physical entities and Transfinite&Elliptic algorithm

[Christophe Geuzaine]

> Dear gmsh,
> I am a gmsh user in China.I just try to learn FEM by it.Now I just
> meet two problems in using gmsh. One is about physical entities.The
> pdf document says,"The purpose of physical entities is to assemble
> elementary entities into larger, possibly overlapping groups, *and
> to control the orientation of the elements in these groups*.If
> physical entities are defined, *the output mesh only contains those
> elements that belong to physical entities.* "This is useful,but I
> can't make it.The orientation(the sequence of the node's num ) is
> the same before and after I define some phisical entities.So is the
> output mesh which contains all the meshing elements I 
> defined-physical or not.Would you please show me how to use
> physical entities effectively,maybe give me an example?

Have a look at tutorial/t1.geo:

- With the file as it is, Gmsh will only save the line elements
discretizing the lines 1, 2 and 4 (as well as two points at vertices 1
and 2, and the triangles discretizing the surface 6).

- Now, if you replace the line

Physical Line(10) = {1,2,4} ;

with

Physical Line(10) = {1,-2,4} ;

you will see that the line elements discretizing the line 2 will be
saved with their orientation reversed.

- In the same vein, it you replace

Physical Surface(MySurface) = {6} ;

with

Physical Surface(MySurface) = {-6} ;

you will see that the triangles will have their normals reversed.

- And if you remove all the "Physical" definitions, even the elements
belonging to line 3 will be saved (there were not going to be saved
using the original file, since line 3 did not belong to any physical
entity).


> The other is about *Transfinite&Elliptic algorithm*.Would you please
> give me a brief introduction about these two algorithm?
> I would appreciate if you reply.

         v

         |
       s4|     c3    s3
         *-----------*
         |           |
       c4|           |c2
         |           |
         *-----------*----- u
        s1     c1    s2

For the quadrangle pictured above, transfinite interpolation is defined
as follows: Given the values c1(u), c2(v), c3(u) and c4(v) on the four
sides (and the values s1, s2, s3 and s4 at the four corners), the
transfinite interpolation f(u,v) of these values at the interior point
(u,v) is

f(u,v) = (1-u)c4(v) + u c2(v) + (1-v)c1(u) + v c3(u)
       - [ (1-u)(1-v)s1 + u(1-v)s2 + uv s3 + (1-u)v s4 ]

The transfinite mesh algorithm uses such an interpolation to determine
the location of interior vertices, given the location of the vertices on
its boundary. (The same kind of interpolation can also be defined for
triangles, hexahedra or prisms.)

The elliptic mesh algorithm is somewhat similar, but instead of using
transfinite interpolation to place the interior vertices, it solves an
elliptic PDE. The goal is to produce "smoother", quasi-orthogonal meshes.

Christophe

-- 
Christophe Geuzaine
Applied and Computational Mathematics, Caltech
geuzaine at acm.caltech.edu - http://geuz.org