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<DIV>Dear Jean-Francois,</DIV>
<DIV> </DIV>
<DIV>Thank you very much for your help.</DIV>
<DIV> </DIV>
<DIV><STRONG><FONT color=#008000>Jean-Francois wrote></FONT></STRONG></DIV>
<DIV><FONT color=#008000>So, when you draw, let's say one quadrangle, you first
split it into 2 triangles and use your quadrilateral interpolation at points,
Then, if your solution is higher order, you split recursively the triangles into
4 triangles, the recursion stops when your solution is sufficiently linear on
the actual triangle.</FONT></DIV>
<DIV><FONT color=#008000></FONT> </DIV>
<DIV>Actually, I will first divide my 4-node (black nodes) quadrangular finite
element into four 4-node quadrangles (figure 1) with five new nodes (white
nodes) in which I will calculate the pressure using 4-node quadrangular finite
element interpolation.</DIV>
<DIV><IMG alt="" hspace=0 src="cid:00dc01c0abdc$94bad180$9e7581c1@bhead.co.uk"
align=baseline border=0></DIV>
<DIV><STRONG>Figure 1: </STRONG>4-node quadrangular finite element with four new
4-node quadrangles.</DIV>
<DIV> </DIV>
<DIV>Then, I will calculate the difference
between linear and 4-node quadrangular finite element
interpolated pressure values in red nodes (see figure 2).</DIV>
<DIV><IMG alt="" hspace=0 src="cid:00dd01c0abdc$94bad180$9e7581c1@bhead.co.uk"
align=baseline border=0></DIV>
<DIV><STRONG>Figure 2: </STRONG>4-node quadrangular finite element with four new
4-node quadrangles. Red nodes are nodes in which we calculate the difference
between linear and 4-node quadrangular finite element
interpolated pressure values.</DIV>
<DIV> </DIV>
<DIV>
<DIV>If the difference (for all red nodes in figure 2) is not small enough (the
difference will be compared to the pressure change on the element) the
subdivision should continue as shown in figure 3, where again we calculate
the difference between linear and 4-node quadrangular finite
element interpolated pressure values in red nodes.</DIV></DIV>
<DIV><IMG alt="" hspace=0 src="cid:00de01c0abdc$94bc5820$9e7581c1@bhead.co.uk"
align=baseline border=0></DIV>
<DIV><STRONG>Figure 3: </STRONG>4-node quadrangular finite element with sixteen
new 4-node quadrangles. Red nodes are nodes in which we calculate the difference
between linear and 4-node quadrangular finite element
interpolated pressure values.</DIV>
<DIV> </DIV>
<DIV>If the difference (for all red nodes) is small enough (say, that it was in
figure 3) there is no need for further subdivisions and we are going
back to a previous subdivision and form 3-node triangles as shown in figure
4.</DIV>
<DIV><IMG alt="" hspace=0 src="cid:00df01c0abdc$94bc5820$9e7581c1@bhead.co.uk"
align=baseline border=0></DIV>
<DIV>
<DIV><STRONG>Figure 4: </STRONG>Final triangulation of 4-node quadrangular
finite element. White nodes are nodes in which we calculate 4-node
quadrangular finite element interpolated pressure values.</DIV>
<DIV> </DIV>
<DIV>I intend to write an ANSI C program which will for every accept
scalar-quadrangle-value in the input ASCII file generate a
corresponding scalar-triangle-value-s in the ASCII output file.</DIV>
<DIV> </DIV>
<DIV>I do not know when I will start working on this, but if I will ever
and if I finish the program I will pass the C source to
you.</DIV>
<DIV> </DIV>
<DIV>Nevertheless in my opinion a proper way to do it is to implement the
support to scalar-quadrangle-value into Gmsh (imagine that you want to zoom
something to extremes, etc.).</DIV>
<DIV> </DIV>
<DIV>Thank you.</DIV>
<DIV> </DIV>
<DIV>Yours sincerely,</DIV>
<DIV> </DIV>
<DIV>Robert Jurjevic</DIV></DIV>
<DIV> </DIV>
<DIV>----- Original Message -----
<DIV>From: <<A
href="mailto:remacle@scorec.rpi.edu">remacle@scorec.rpi.edu</A>></DIV>
<DIV>To: "Robert Jurjevic" <<A
href="mailto:robert.jurjevic@bhead.co.uk">robert.jurjevic@bhead.co.uk</A>></DIV>
<DIV>Cc: "christophe geuzaine" <<A
href="mailto:Christophe.Geuzaine@ulg.ac.be">Christophe.Geuzaine@ulg.ac.be</A>></DIV>
<DIV>Sent: Tuesday, March 13, 2001 2:18 PM</DIV>
<DIV>Subject: Re: scalar-quadrangle-value in Gmsh's ASCII post-processing
file</DIV></DIV>
<DIV><BR></DIV>> Hello Robert,<BR>> <BR>> ><BR>> > Thank you
for developing Gmsh!<BR>> ><BR>> <BR>> You're welcome.<BR>>
<BR>> > My name is Robert Jurjevic and some time ago I wrote a solver for
a<BR>> > classical fluid mechanics system of partial differential
equations modelling<BR>> > two-dimensional stationary incompressible
Newtonian fluid flow.<BR>> ><BR>> > I tried the solver on classical
examples of driven cavity laminar flow,<BR>> > laminar flow past a
cylinder and a laminar flow past a symmetric NACA<BR>> > airfoil.<BR>>
><BR>> > I used 2D 8-node superparametric (pressure) and isoparametric
(velocity)<BR>> > quadrilateral finite elements.<BR>> ><BR>> >
Pre- and post-processing has been handled using I-DEAS Finite Element<BR>>
> Modelling software running on an HP UNIX workstation.<BR>> ><BR>>
> Now when there is a Gmsh, I thought that I could produce a coloured
picture<BR>> > of a NACA airfoil pressure field on my PC.<BR>>
><BR>> > I kindly beg you if you could answer the following questions.
Thank you.<BR>> ><BR>> <BR>> Yes, I use here high order finite
elements for compressible 3d flows, gmsh<BR>> can display everything but the
only primitives are simplices (points,<BR>> lines, triangles,
tetrahedron).<BR>> <BR>> For displaying higher order<BR>>
interpolations, we use a recursive algorithm. First, you must undesrtand<BR>>
that triangles that you display in gmsh are not s mesh, they may be<BR>>
completely disconnected even overlapping (I do not say that it's<BR>>
recommended). So, when you draw, le's say one quandrangles, you first<BR>>
split it ito 2 triangles and use your quadrilateral interpolant at<BR>>
points<BR>> <BR>> (-1,-1), (-1,1), (1,1) for the first triangle<BR>>
(1,1) , (1,-1), (-1,-1) for the second triangle<BR>> <BR>> Then, if your
solution is higher order, you split recursively the<BR>> triangles into 4
triangles, the recursion stops when your solution is<BR>> sufficiently linear
on the actual triangle. Look at the attachement, this<BR>> is a very
classical example of a supersonic flow (splitted-quads.gif). The<BR>> image
shows the subdivision.<BR>> <BR>> In the other example (iso.jpg), I
splitted hexaedron into tetrahedron,<BR>> then recurse.<BR>> <BR>>
<BR>> > QUESTIONS<BR>> ><BR>> > 1. Is there a
scalar-quadrangle-value equivalent of the<BR>> > scalar-triangle-value in
the Gmsh ASCII post-processing file. The pressure<BR>> > in my examples is
calculated using 2D 8-node superparametric quadrilateral<BR>> > finite
elements, but approximation with 2D 4-node isoparametric<BR>> >
quadrilateral finite elements will do fine. I know that I can approximate
a<BR>> > 2D 4-node isoparametric quadrangle finite element with two 2D
3-node<BR>> > isoparametric triangular finite elements, but I would like
to avoid that<BR>> > (extra work and the error).<BR>> <BR>> No, look
below.<BR>> <BR>> ><BR>> > 2. Do you intend to implement the
support for 2D 8-node isoparametric finite<BR>> > elements (for both pre-
and post-processing)? If you introduce something<BR>> > like
vector-quadrilateral8-value I would be able to present the velocity<BR>> >
field (variable length arrows i presume).<BR>> ><BR>> <BR>> <BR>>
No.<BR>> <BR>> > If I manage to produce the colourful picture of
pressure field for the flow<BR>> > around the airfoil I will send it (the
picture or the data to generate the<BR>> > picture) to you.<BR>>
><BR>> <BR>> Thank you very much. Moreover, we are actually developping
plugin<BR>> capabilities for gmsh, if you want to plug your software on gmsh,
we can<BR>> give you the material (i.e. source code of the core).<BR>>
<BR>> Jean-Francois.<BR>> <BR>> <BR>> > In fact the sole purpose
of producing the pressure picture is due to its<BR>> > aesthetic value and
I regard it as a peace of scientific art.<BR>> ><BR>> > Yours
sincerely,<BR>> ><BR>> > Robert Jurjevic<BR>> ><BR>>
><BR>>
<P>
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<P></P>> [Email attachment "iso.jpg" ("IMAGE/jpeg") denied by Firewall]
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