1. README
2. t1.geo
3. t2.geo
4. t3.geo
5. t4.geo
6. t5.geo
7. t6.geo
8. t7.geo
9. t8.geo
10. t9.geo

$Id: tutorial.html,v 1.24 2001-10-10 07:15:22 geuzaine Exp $

Here are the examples in the Gmsh tutorial. These examples are
commented (both C and C++-style comments can be used in Gmsh input
files) and should introduce new features gradually, starting with
t1.geo.

[NOTE: This tutorial does not explain the mesh and post-processing
file formats. See the FORMATS file for this.]

There are two ways to actually run these examples with Gmsh. (The
operations to run Gmsh may vary according to your operating system. In
the folowing examples, we will assume that you're working with a
UNIX-like shell.) The first working mode of Gmsh is the interactive
graphical mode. To launch Gmsh in interactive mode, just type

> gmsh

at the prompt on the command line. This will open two windows: the
graphic window (with a status bar at the bottom) and the menu window
(with a menu bar and some context dependent buttons). To open the
first tutorial file, select the 'File->Open' menu, and choose 't1.geo'
in the input field. To perform the mesh generation, go to the mesh
module (by selecting 'Mesh' in the module menu) and choose the
required dimension in the context-dependent buttons ('1D' will mesh
all the curves; '2D' will mesh all the surfaces ---as well as all the
curves if '1D' was not called before; '3D' will mesh all the volumes
---and all the surfaces if '2D' was not called before). To save the
resulting mesh in the current mesh format, choose 'Save' in the
context-dependent buttons, or select the appropriate format with the
'File->Save as' menu. The default mesh file name is based on the name
of the first input file on the command line (or 'unnamed' if there
wasn't any input file given), with an appended extension depending on
the mesh format.

[NOTE: Nearly all the interactive commands have shortcuts. Select
'Help->Shortcuts' in the menu bar to learn about these shortcuts.]

Instead of opening the tutorial with the 'File->Open' menu, it is
often more convenient to put the file name on the command line, for
example with:

> gmsh t1.geo

[NOTE: The '.geo' extension can also be omitted.]

[NOTE: Even if it is often handy to define the variables and the
points directly in the input files (you may use any text editor for
this purpose, e.g. Wordpad on Windows, or Emacs on Unix), it is almost
always more simple to define the curves, the surfaces and the volumes
interactively. To do so, just follow the context dependent buttons in
the Geometry module. For example, to create a spline, select
'Geometry' in the module menu, and then select 'Elementary, Add, New,
Spline'. You will then be asked (in the status bar of the graphic
window) to select a list of points, and to click 'e' to finish the
selection (or 'q' to abort it). Once the interactive command is
completed, a string is automatically added at the end of the currently
opened project file.]

The second operating mode for Gmsh is the non-interactive mode. In
this mode, there is no graphical user interface, and all operations
are performed without any interaction. To mesh the first tutorial in
non-interactive mode, just type:

> gmsh t1.geo -2

To mesh the same example, but with the backgound mesh available in the
file 'bgmesh.pos', just type:

> gmsh t1.geo -2 -bgm bgmesh.pos

[NOTE: You should read the notes in the file 'bgmesh.pos' if you
intend to use background meshes.]

Several files can be loaded simultaneously in Gmsh. The first one
defines the project, while the others are appended ("merged") to this
project. You can merge such files with the 'File->Merge' menu, or by
directly specifying the names of the files on the command line. This
is most useful for post-processing purposes. For example, to merge the
post-processing views contained in the files 'view1.pos' and
'view2.pos' together with the first tutorial 't1.geo', you can type
the following command:

> gmsh t1.geo view1.pos view2.pos

In the Post-Processing module (select 'Post_Processing' in the module
menu), two view buttons will appear, respectively labeled "a scalar
map" and "a vector map". A left mouse click toggles the visibility of
the selected view. A right mouse click provides access to the view's
options. If you want the modifications made to one view to affect also
all the other views, select the 'Apply next changes to all views' or
'Force same options for all views' option in the
'Options->Post-processing' menu.

[NOTE: All the options specified interactively can also be directly
specified in the ascii input files. All available options, with their
current values, can be saved into a file by selecting 'File->Save
as->GEO complete options', or simply viewed by pressing the '?' button
in the status bar. To save the current options as your default
preferences for all future Gmsh sessions, use the 'Options->Save
options now' menu.]


OK, that's all, folks. Enjoy the tutorial.

/********************************************************************* 
 *
 *  Gmsh tutorial 1
 * 
 *  Variables, Elementary entities (Points, Lines, Surfaces), Physical
 *  entities (Points, Lines, Surfaces), Background mesh
 *
 *********************************************************************/

// All geometry description in Gmsh is made by means of a special
// language (looking somewhat similar to C). The simplest construction
// of this language is the 'affectation'. 

// The following command (all commands end with a semi colon) defines
// a variable called 'lc' and affects the value 0.007 to 'lc':

lc = 0.007 ;

// This newly created variable can be used to define the first Gmsh
// elementary entity, a 'Point'. A Point is defined by a list of four
// numbers: its three coordinates (x, y and z), and a characteristic
// length which sets the target mesh size at the point:

Point(1) = {0,  0,  0, 9.e-1 * lc} ;

// The mesh size is defined as the length of the segments for lines,
// the radii of the circumscribed circles for triangles and the radii
// of the circumscribed spheres for tetrahedra, respectively. The
// actual distribution of the mesh sizes is obtained by interpolation
// of the characteristic lengths prescribed at the points. There are
// also other possibilities to specify characteristic lengths:
// attractors (see t7.geo) and background meshes (see bgmesh.pos).

// As can be seen in the previous definition, more complex expressions
// can be constructed from variables. Here, the product of the
// variable 'lc' by the constant 9.e-1 is given as the fourth argument
// of the list defining the point.
//
// The following general syntax rule is applied for the definition of
// all geometrical entities:
//
//    "If a number defines a new entity, it is enclosed between
//    parentheses. If a number refers to a previously defined entity,
//    it is enclosed between braces."
//
// Three additional points are then defined:

Point(2) = {.1, 0,  0, lc} ;
Point(3) = {.1, .3, 0, lc} ;
Point(4) = {0,  .3, 0, lc} ;

// The second elementary geometrical entity in Gmsh is the
// curve. Amongst curves, straight lines are the simplest. A straight
// line is defined by a list of point numbers. For example, line 1
// starts at point 1 and ends at point 2:

Line(1) = {1,2} ;
Line(2) = {3,2} ;
Line(3) = {3,4} ;
Line(4) = {4,1} ;

// The third elementary entity is the surface. In order to define a
// simple rectangular surface from the four lines defined above, a
// line loop has first to be defined. A line loop is a list of
// connected lines, a sign being associated with each line (depending
// on the orientation of the line).

Line Loop(5) = {4,1,-2,3} ;

// The surface is then defined as a list of line loops (only one
// here):

Plane Surface(6) = {5} ;

// At this level, Gmsh knows everything to display the rectangular
// surface 6 and to mesh it. But a supplementary step is needed in
// order to assign region numbers to the various elements in the mesh
// (the points, the lines and the triangles discretizing points 1 to
// 4, lines 1 to 4 and surface 6). This is achieved by the definition
// of Physical entities. Physical entities will group elements
// belonging to several elementary entities by giving them a common
// number (a region number), and specifying their orientation.
//
// For example, the two points 1 and 2 can be grouped into the
// physical entity 1:

Physical Point(1) = {1,2} ;

// Consequently, two punctual elements will be saved in the output
// files, both with the region number 1. The mechanism is identical
// for line or surface elements:

Physical Line(10) = {1,2,4} ;
Physical Surface(100) = {6} ;

// All the line elements which will be created during the mesh of
// lines 1, 2 and 4 will be saved in the output file with the region
// number 10; and all the triangular elements resulting from the
// discretization of surface 6 will be given the region number 100.
//
// If no physical groups are defined, all the elements in the mesh are
// directly saved with their default orientation and with a region
// number equal to their elementary region number. For a description
// of the mesh and post-processing formats, see the FORMATS file.

/********************************************************************* 
 *
 *  Gmsh tutorial 2
 * 
 *  Includes, Geometrical transformations, Extruded geometries,
 *  Elementary entities (Volumes), Physical entities (Volumes)
 *
 *********************************************************************/

// The first tutorial file will serve as a basis to construct this
// one. It can be included with:

Include "t1.geo" ;

// There are several possibilities to build a more complex geometry
// from the one previously defined in 't1.geo'.
//
// New points, lines and surfaces can first be directly defined in the
// same way as in 't1.geo':

Point(5) = {0, .4, 0, lc} ;
Line(5) = {4, 5} ;

// But Gmsh also provides geometrical transformation mechanisms to
// move (translate, rotate, ...), add (translate, rotate, ...) or
// extrude (translate, rotate) elementary geometrical entities. For
// example, the point 3 can be moved by 0.05 units on the left with:

Translate {-0.05,0,0} { Point{3} ; }

// The resulting point can also be duplicated and translated by 0.1
// along the y axis:

Translate {0,0.1,0} { Duplicata{ Point{3} ; } }

// Of course, translation, rotation and extrusion commands not only
// apply to points, but also to lines and surfaces. The following
// command extrudes surface 6 defined in 't1.geo', as well as a new
// surface 11, along the z axis by 'h':

h = 0.12 ;
Extrude Surface { 6, {0, 0, h} } ;

Line(7) = {3, 6} ; Line(8) = {6,5} ; Line Loop(10) = {5,-8,-7,3};

Plane Surface(11) = {10};

Extrude Surface { 11, {0, 0, h} } ;

// All these geometrical transformations automatically generate new
// elementary entities. The following commands permit to specify
// manually a characteristic length for some of the automatically
// created points:

Characteristic Length{6,22,2,3,16,12} = lc * 3 ;

// If the transformation tools are handy to create complex geometries,
// it is sometimes useful to generate the flat geometry, consisting
// only of the explicit list elementary entities. This can be achieved
// by selecting the 'File->Save as->GEO flattened geometry' menu or 
// by typing
//
// > gmsh t2.geo -0
//
// on the command line.

// Volumes are the fourth type of elementary entities in Gmsh. In the
// same way one defines line loops to build surfaces, one has to
// define surface loops to build volumes. The following volumes are
// very simple, without holes (and thus consist of only one surface
// loop):

Surface Loop(145) = {121,11,131,135,139,144};
Volume(146) = {145};

Surface Loop(146) = {121,6,109,113,117,122};
Volume(147) = {146};

// To save all volumic (tetrahedral) elements of volume 146 and 147
// with the associate region number 1, a Physical Volume must be
// defined:

Physical Volume (1) = {146,147} ;

// Congratulations! You've created your first fully unstructured
// tetrahedral 3D mesh!

/********************************************************************* 
 *
 *  Gmsh tutorial 3
 * 
 *  Extruded meshes, Options
 *
 *********************************************************************/

// Again, the first tutorial example is included:

Include "t1.geo" ;

// As in 't2.geo', an extrusion along the z axis will be performed:

h = 0.1 ;

// But contrary to 't2.geo', not only the geometry will be extruded,
// but also the 2D mesh. This is done with the same Extrude command,
// but by specifying the number of layers (here, there will be four
// layers, of respectively 8, 4, 2 and 1 elements in depth), with
// volume numbers 9000 to 9003 and respective heights equal to h/4:

Extrude Surface { 6, {0,0,h} } { 
  Layers { {8,4,2,1}, {9000:9003}, {0.25,0.5,0.75,1} } ; 
} ;

// The extrusion can also be combined with a rotation, and the
// extruded 3D mesh can be recombined into prisms (wedges) if the
// surface elements are triangles, or hexahedra if the surface
// elements are quadrangles. All rotations are specified by an axis
// direction ({0,1,0}), an axis point ({-0.1,0,0.1}) and a rotation
// angle (-Pi/2):

Extrude Surface { 122, {0,1,0} , {-0.1,0,0.1} , -Pi/2 } { 
  Recombine ; Layers { 7, 9004, 1 } ; 
};

Physical Volume(101) = {9000:9004};

// All interactive options can also be set directly in the input file.
// For example, the following lines define a global characteristic
// length factor, redefine some background colors, disable the display
// of the axes, and select an initial viewpoint in XYZ mode (disabling
// the interactive trackball-like rotation mode):

Mesh.CharacteristicLengthFactor = 4;
General.Color.Background = {120,120,120};
General.Color.Foreground = {255,255,255};
General.Color.Text = White;
Geometry.Color.Points = Orange;
General.Axes = 0;
General.Trackball = 0;
General.RotationX = 10;
General.RotationY = 70;
General.TranslationX = -0.1;

// Note: all colors can be defined literally or numerically, i.e.
// 'General.Color.Background = Red' is equivalent to
// 'General.Color.Background = {255,0,0}'. As with user-defined
// variables, the options can be used either as right hand or left
// hand sides, so that

Geometry.Color.Surfaces = Geometry.Color.Points;

// will assign the color of the surfaces in the geometry to the same
// color as the points.

// A click on the '?'  button in the status bar of the graphic window
// will dump all current options to the terminal. To save all
// available options to a file, use the 'File->Save as->GEO complete
// options' menu. To save the current options as the default options
// for all future Gmsh sessions, use the 'Options->Save options now'
// menu.

/********************************************************************* 
 *
 *  Gmsh tutorial 4
 * 
 *  Built-in functions, Holes
 *
 *********************************************************************/

cm = 1e-02 ;

e1 = 4.5*cm ; e2 = 6*cm / 2 ; e3 =  5*cm / 2 ;

h1 = 5*cm ; h2 = 10*cm ; h3 = 5*cm ; h4 = 2*cm ; h5 = 4.5*cm ;

R1 = 1*cm ; R2 = 1.5*cm ; r = 1*cm ;

ccos = ( -h5*R1 + e2 * Hypot(h5,Hypot(e2,R1)) ) / (h5^2 + e2^2) ;
ssin = Sqrt(1-ccos^2) ;

Lc1 = 0.01 ;
Lc2 = 0.003 ;

// A whole set of operators can be used, which can be combined in all
// the expressions. These operators are defined in a similar way to
// their C or C++ equivalents (with the exception of '^'):
//
//   '-' (in both unary and binary versions, i.e. as in '-1' and '1-2')
//   '!' (the negation)
//   '+'
//   '*'
//   '/'
//   '%' (the rest of the integer division)
//   '<'
//   '>'
//   '<='
//   '>='
//   '=='
//   '!='
//   '&&' (and)
//   '||' (or)
//   '||' (or)
//   '^' (power)
//   '?' ':' (the ternary operator)
//
// Grouping is done, as usual, with parentheses.
//
// In addition to these operators, all C mathematical functions can
// also be used (note the first capital letter), i.e.
// 
//   Exp(x)
//   Log(x)
//   Log10(x)
//   Sqrt(x)
//   Sin(x)
//   Asin(x)
//   Cos(x)
//   Acos(x)
//   Tan(x)
//   Atan(x)
//   Atan2(x,y)
//   Sinh(x)
//   Cosh(x)
//   Tanh(x)
//   Fabs(x)
//   Floor(x)
//   Ceil(x)
//   Fmod(x,y)
// 
// as well as a series of other functions:
//
//   Hypot(x,y)   computes Sqrt(x^2+y^2)
//   Rand(x)      generates a random number in [0,x]
//
// The only predefined constant in Gmsh is Pi.

Point(1) = { -e1-e2, 0.0  , 0.0 , Lc1};
Point(2) = { -e1-e2, h1   , 0.0 , Lc1};
Point(3) = { -e3-r , h1   , 0.0 , Lc2};
Point(4) = { -e3-r , h1+r , 0.0 , Lc2};
Point(5) = { -e3   , h1+r , 0.0 , Lc2};
Point(6) = { -e3   , h1+h2, 0.0 , Lc1};
Point(7) = {  e3   , h1+h2, 0.0 , Lc1};
Point(8) = {  e3   , h1+r , 0.0 , Lc2};
Point(9) = {  e3+r , h1+r , 0.0 , Lc2};
Point(10)= {  e3+r , h1   , 0.0 , Lc2};
Point(11)= {  e1+e2, h1   , 0.0 , Lc1};
Point(12)= {  e1+e2, 0.0  , 0.0 , Lc1};
Point(13)= {  e2   , 0.0  , 0.0 , Lc1};

Point(14)= {  R1 / ssin , h5+R1*ccos, 0.0 , Lc2};
Point(15)= {  0.0       , h5        , 0.0 , Lc2};
Point(16)= { -R1 / ssin , h5+R1*ccos, 0.0 , Lc2};
Point(17)= { -e2        , 0.0       , 0.0 , Lc1};

Point(18)= { -R2  , h1+h3   , 0.0 , Lc2};
Point(19)= { -R2  , h1+h3+h4, 0.0 , Lc2};
Point(20)= {  0.0 , h1+h3+h4, 0.0 , Lc2};
Point(21)= {  R2  , h1+h3+h4, 0.0 , Lc2};
Point(22)= {  R2  , h1+h3   , 0.0 , Lc2};
Point(23)= {  0.0 , h1+h3   , 0.0 , Lc2};

Point(24)= {  0 , h1+h3+h4+R2, 0.0 , Lc2};
Point(25)= {  0 , h1+h3-R2,    0.0 , Lc2};

Line(1)  = {1 ,17};
Line(2)  = {17,16};

// All curves are not straight lines... Circles are defined by a list
// of three point numbers, which represent the starting point, the
// center and the end point, respectively. All circles have to be
// defined in the trigonometric (counter-clockwise) sense.  Note that
// the 3 points should not be aligned (otherwise the plane in which
// the circle lies has to be defined, by 'Circle(num) =
// {start,center,end} Plane {nx,ny,nz}').

Circle(3) = {14,15,16};
Line(4)  = {14,13};
Line(5)  = {13,12};
Line(6)  = {12,11};
Line(7)  = {11,10};
Circle(8) = { 8, 9,10};
Line(9)  = { 8, 7};
Line(10) = { 7, 6};
Line(11) = { 6, 5};
Circle(12) = { 3, 4, 5};
Line(13) = { 3, 2};
Line(14) = { 2, 1};
Line(15) = {18,19};
Circle(16) = {21,20,24};
Circle(17) = {24,20,19};
Circle(18) = {18,23,25};
Circle(19) = {25,23,22};
Line(20) = {21,22};

Line Loop(21) = {17,-15,18,19,-20,16};
Plane Surface(22) = {21};

// The surface is made of two line loops, i.e. it has one hole:

Line Loop(23) = {11,-12,13,14,1,2,-3,4,5,6,7,-8,9,10};
Plane Surface(24) = {23,21};

Physical Surface(1) = {22};
Physical Surface(2) = {24};

/********************************************************************* 
 *
 *  Gmsh tutorial 5
 * 
 *  Characteristic lengths, Arrays of variables, Functions, Loops
 *
 *********************************************************************/

// This defines some characteristic lengths:

lcar1 = .1;
lcar2 = .0005;
lcar3 = .075;

// In order to change these lengths globally (without changing the
// file), a global scaling factor for all characteristic lengths can
// be specified on the command line with the option '-clscale' (or
// with the option Mesh.CharacteristicLengthFactor). For example,
// with:
//
// > gmsh t5 -clscale 1
//
// this example produces a mesh of approximately 2000 nodes and
// 10,000 tetrahedra (in 3 seconds on an alpha workstation running at
// 666MHz). With 
//
// > gmsh t5 -clscale 0.2
//
// (i.e. with all characteristic lengths divided by 5), the mesh
// counts approximately 170,000 nodes and one million tetrahedra
// (and the computation takes 16 minutes on the same machine :-( So
// there is still a lot of work to do to achieve decent performance
// with Gmsh...)

Point(1) = {0.5,0.5,0.5,lcar2}; Point(2) = {0.5,0.5,0,lcar1};
Point(3) = {0,0.5,0.5,lcar1};   Point(4) = {0,0,0.5,lcar1}; 
Point(5) = {0.5,0,0.5,lcar1};   Point(6) = {0.5,0,0,lcar1};
Point(7) = {0,0.5,0,lcar1};     Point(8) = {0,1,0,lcar1};
Point(9) = {1,1,0,lcar1};       Point(10) = {0,0,1,lcar1};
Point(11) = {0,1,1,lcar1};      Point(12) = {1,1,1,lcar1};
Point(13) = {1,0,1,lcar1};      Point(14) = {1,0,0,lcar1};

Line(1) = {8,9};    Line(2) = {9,12};  Line(3) = {12,11};
Line(4) = {11,8};   Line(5) = {9,14};  Line(6) = {14,13};
Line(7) = {13,12};  Line(8) = {11,10}; Line(9) = {10,13};
Line(10) = {10,4};  Line(11) = {4,5};  Line(12) = {5,6};
Line(13) = {6,2};   Line(14) = {2,1};  Line(15) = {1,3};
Line(16) = {3,7};   Line(17) = {7,2};  Line(18) = {3,4};
Line(19) = {5,1};   Line(20) = {7,8};  Line(21) = {6,14};

Line Loop(22) = {11,19,15,18};       Plane Surface(23) = {22};
Line Loop(24) = {16,17,14,15};       Plane Surface(25) = {24};
Line Loop(26) = {-17,20,1,5,-21,13}; Plane Surface(27) = {26};
Line Loop(28) = {4,1,2,3};           Plane Surface(29) = {28};
Line Loop(30) = {7,-2,5,6};          Plane Surface(31) = {30};
Line Loop(32) = {6,-9,10,11,12,21};  Plane Surface(33) = {32};
Line Loop(34) = {7,3,8,9};           Plane Surface(35) = {34};
Line Loop(36) = {10,-18,16,20,-4,8}; Plane Surface(37) = {36};
Line Loop(38) = {-14,-13,-12,19};    Plane Surface(39) = {38};

// Instead of using included files, one can also define functions. In
// the following function, the reserved variable 'newp' is used, which
// automatically selects a new point number. This number is chosen as
// the highest current point number, plus one. Analogously to 'newp',
// there exists a variable 'newreg' which selects the highest number
// of all entities other than points, plus one.

// Note: there are no local variables. This will be changed in a
// future version of Gmsh.

Function CheeseHole 

  p1 = newp; Point(p1) = {x,  y,  z,  lcar3} ;
  p2 = newp; Point(p2) = {x+r,y,  z,  lcar3} ;
  p3 = newp; Point(p3) = {x,  y+r,z,  lcar3} ;
  p4 = newp; Point(p4) = {x,  y,  z+r,lcar3} ;
  p5 = newp; Point(p5) = {x-r,y,  z,  lcar3} ;
  p6 = newp; Point(p6) = {x,  y-r,z,  lcar3} ;
  p7 = newp; Point(p7) = {x,  y,  z-r,lcar3} ;

  c1 = newreg; Circle(c1) = {p2,p1,p7};
  c2 = newreg; Circle(c2) = {p7,p1,p5};
  c3 = newreg; Circle(c3) = {p5,p1,p4};
  c4 = newreg; Circle(c4) = {p4,p1,p2};
  c5 = newreg; Circle(c5) = {p2,p1,p3};
  c6 = newreg; Circle(c6) = {p3,p1,p5};
  c7 = newreg; Circle(c7) = {p5,p1,p6};
  c8 = newreg; Circle(c8) = {p6,p1,p2};
  c9 = newreg; Circle(c9) = {p7,p1,p3};
  c10 = newreg; Circle(c10) = {p3,p1,p4};
  c11 = newreg; Circle(c11) = {p4,p1,p6};
  c12 = newreg; Circle(c12) = {p6,p1,p7};

// All surfaces are not plane... Here is the way to define ruled
// surfaces (which have 3 or 4 borders):

  l1 = newreg; Line Loop(l1) = {c5,c10,c4};   Ruled Surface(newreg) = {l1};
  l2 = newreg; Line Loop(l2) = {c9,-c5,c1};   Ruled Surface(newreg) = {l2};
  l3 = newreg; Line Loop(l3) = {-c12,c8,c1};  Ruled Surface(newreg) = {l3};
  l4 = newreg; Line Loop(l4) = {c8,-c4,c11};  Ruled Surface(newreg) = {l4};
  l5 = newreg; Line Loop(l5) = {-c10,c6,c3};  Ruled Surface(newreg) = {l5};
  l6 = newreg; Line Loop(l6) = {-c11,-c3,c7}; Ruled Surface(newreg) = {l6};
  l7 = newreg; Line Loop(l7) = {c2,c7,c12};   Ruled Surface(newreg) = {l7};
  l8 = newreg; Line Loop(l8) = {-c6,-c9,c2};  Ruled Surface(newreg) = {l8};

// Warning: surface meshes are generated by projecting a 2D mesh in
// the mean plane of the surface. This gives nice results only if the
// surface curvature is small enough. Otherwise you will have to cut
// the surface in pieces.

// Arrays of variables can be manipulated in the same way as classical
// variables. Warning: accessing an uninitialized element in an array
// will produce an unpredictable result. Note that whole arrays can
// also be instantly initialized (e.g. l[]={1,2,7} is valid).

  theloops[t] = newreg ; 

  Surface Loop(theloops[t]) = {l8+1, l5+1, l1+1, l2+1, -(l3+1), -(l7+1), l6+1, l4+1};

  thehole = newreg ; 
  Volume(thehole) = theloops[t] ;

Return


x = 0 ; y = 0.75 ; z = 0 ; r = 0.09 ;

// A For loop is used to generate five holes in the cube:

For t In {1:5}

  x += 0.166 ; 
  z += 0.166 ; 

// This command calls the function CheeseHole. Note that, instead of
// defining a function, we could have defined a file containing the
// same code, and used the Include command to include this file.

  Call CheeseHole ;

// A physical volume is defined for each cheese hole

  Physical Volume (t) = thehole ;
 
// The Printf function permits to print the value of variables on the
// terminal, in a way similar to the 'printf' C function:

  Printf("The cheese hole %g (center = {%g,%g,%g}, radius = %g) has number %g!",
	 t, x, y, z, r, thehole) ;

// Note: All Gmsh variables are treated internally as double precision
// numbers. The format string should thus only contain valid double
// precision number format specifiers (see the C or C++ language
// reference for more details).

EndFor

// This is the surface loop for the exterior surface of the cube:

theloops[0] = newreg ;

Surface Loop(theloops[0]) = {35,31,29,37,33,23,39,25,27} ;

// The volume of the cube, without the 5 cheese holes, is defined by 6
// surface loops (the exterior surface and the five interior loops).
// To reference an array of variables, its identifier is followed by
// '[]':

Volume(186) = {theloops[]} ;

// This physical volume assigns the region number 10 to the tetrahedra
// paving the cube (but not the holes, whose elements were tagged from
// 1 to 5 in the 'For' loop)

Physical Volume (10) = 186 ;

/********************************************************************* 
 *
 *  Gmsh tutorial 6
 * 
 *  Transfinite meshes
 *
 *********************************************************************/

r_int  = 0.05 ;
r_ext  = 0.051 ;
r_far  = 0.125 ;
r_inf  = 0.4 ;
phi1   = 30. * (Pi/180.) ;
angl   = 45. * (Pi/180.) ;

nbpt_phi   = 5 ; nbpt_int   = 20 ;
nbpt_arc1  = 10 ; nbpt_arc2  = 10 ;
nbpt_shell = 10 ; nbpt_far   = 25 ; nbpt_inf = 15 ;

lc0 = 0.1 ; lc1 = 0.1 ; lc2 = 0.3 ;

Point(1) = {0,     0, 0, lc0} ;
Point(2) = {r_int, 0, 0, lc0} ;
Point(3) = {r_ext, 0, 0, lc1} ;
Point(4) = {r_far, 0, 0, lc2} ;
Point(5) = {r_inf, 0, 0, lc2} ;
Point(6) = {0, 0,  r_int, lc0} ;
Point(7) = {0, 0,  r_ext, lc1} ;
Point(8) = {0, 0,  r_far, lc2} ;
Point(9) = {0, 0,  r_inf, lc2} ;

Point(10) = {r_int*Cos(phi1), r_int*Sin(phi1), 0, lc0} ;
Point(11) = {r_ext*Cos(phi1), r_ext*Sin(phi1), 0, lc1} ;
Point(12) = {r_far*Cos(phi1), r_far*Sin(phi1), 0, lc2} ;
Point(13) = {r_inf*Cos(phi1), r_inf*Sin(phi1), 0, lc2} ;

Point(14) = {r_int/2,           0,   0,               lc2} ;
Point(15) = {r_int/2*Cos(phi1), r_int/2*Sin(phi1), 0, lc2} ;
Point(16) = {r_int/2,           0,                 r_int/2, lc2} ;
Point(17) = {r_int/2*Cos(phi1), r_int/2*Sin(phi1), r_int/2, lc2} ;
Point(18) = {0, 0,  r_int/2, lc2} ;
Point(19) = {r_int*Cos(angl),           0,                         r_int*Sin(angl), lc2} ;
Point(20) = {r_int*Cos(angl)*Cos(phi1), r_int*Cos(angl)*Sin(phi1), r_int*Sin(angl), lc2} ;
Point(21) = {r_ext*Cos(angl),           0,                         r_ext*Sin(angl), lc2} ;
Point(22) = {r_ext*Cos(angl)*Cos(phi1), r_ext*Cos(angl)*Sin(phi1), r_ext*Sin(angl), lc2} ;
Point(23) = {r_far*Cos(angl),           0,                         r_far*Sin(angl), lc2} ;
Point(24) = {r_far*Cos(angl)*Cos(phi1), r_far*Cos(angl)*Sin(phi1), r_far*Sin(angl), lc2} ;
Point(25) = {r_inf,           0,                r_inf, lc2} ;
Point(26) = {r_inf*Cos(phi1), r_inf*Sin(phi1),  r_inf, lc2} ;

Circle(1) = {2,1,19};   Circle(2) = {19,1,6};   Circle(3) = {3,1,21};
Circle(4) = {21,1,7};   Circle(5) = {4,1,23};   Circle(6) = {23,1,8};   
Line(7) = {5,25};   Line(8) = {25,9};
Circle(9) = {10,1,20};  Circle(10) = {20,1,6};  Circle(11) = {11,1,22};
Circle(12) = {22,1,7};  Circle(13) = {12,1,24}; Circle(14) = {24,1,8};
Line(15) = {13,26}; Line(16) = {26,9};
Circle(17) = {19,1,20}; Circle(18) = {21,1,22}; Circle(19) = {23,1,24};
Circle(20) = {25,1,26}; Circle(21) = {2,1,10};  Circle(22) = {3,1,11};  
Circle(23) = {4,1,12};  Circle(24) = {5,1,13};

Line(25) = {1,14};  Line(26) = {14,2};  Line(27) = {2,3};   Line(28) = {3,4};
Line(29) = {4,5};   Line(30) = {1,15};  Line(31) = {15,10}; Line(32) = {10,11};
Line(33) = {11,12}; Line(34) = {12,13}; Line(35) = {14,15}; Line(36) = {14,16};
Line(37) = {15,17}; Line(38) = {16,17}; Line(39) = {18,16}; Line(40) = {18,17};
Line(41) = {1,18};  Line(42) = {18,6};  Line(43) = {6,7};   Line(44) = {16,19};
Line(45) = {19,21}; Line(46) = {21,23}; Line(47) = {23,25}; Line(48) = {17,20};
Line(49) = {20,22}; Line(50) = {22,24}; Line(51) = {24,26}; Line(52) = {7,8};
Line(53) = {8,9};

Line Loop(54) = {39,-36,-25,41};  Ruled Surface(55) = {54};
Line Loop(56) = {44,-1,-26,36};   Ruled Surface(57) = {56};
Line Loop(58) = {3,-45,-1,27};    Ruled Surface(59) = {58};
Line Loop(60) = {5,-46,-3,28};    Ruled Surface(61) = {60};
Line Loop(62) = {7,-47,-5,29};    Ruled Surface(63) = {62};
Line Loop(64) = {-2,-44,-39,42};  Ruled Surface(65) = {64};
Line Loop(66) = {-4,-45,2,43};    Ruled Surface(67) = {66};
Line Loop(68) = {-6,-46,4,52};    Ruled Surface(69) = {68};
Line Loop(70) = {-8,-47,6,53};    Ruled Surface(71) = {70};
Line Loop(72) = {-40,-41,30,37};  Ruled Surface(73) = {72};
Line Loop(74) = {48,-9,-31,37};   Ruled Surface(75) = {74};
Line Loop(76) = {49,-11,-32,9};   Ruled Surface(77) = {76};
Line Loop(78) = {-50,-11,33,13};  Ruled Surface(79) = {78};
Line Loop(80) = {-51,-13,34,15};  Ruled Surface(81) = {80};
Line Loop(82) = {10,-42,40,48};   Ruled Surface(83) = {82};
Line Loop(84) = {12,-43,-10,49};  Ruled Surface(85) = {84};
Line Loop(86) = {14,-52,-12,50};  Ruled Surface(87) = {86};
Line Loop(88) = {16,-53,-14,51};  Ruled Surface(89) = {88};
Line Loop(90) = {-30,25,35};      Ruled Surface(91) = {90};
Line Loop(92) = {-40,39,38};      Ruled Surface(93) = {92};
Line Loop(94) = {37,-38,-36,35};  Ruled Surface(95) = {94};
Line Loop(96) = {-48,-38,44,17};  Ruled Surface(97) = {96};
Line Loop(98) = {18,-49,-17,45};  Ruled Surface(99) = {98};
Line Loop(100) = {19,-50,-18,46}; Ruled Surface(101) = {100};
Line Loop(102) = {20,-51,-19,47}; Ruled Surface(103) = {102};
Line Loop(104) = {-2,17,10};      Ruled Surface(105) = {104};
Line Loop(106) = {-9,-21,1,17};   Ruled Surface(107) = {106};
Line Loop(108) = {-4,18,12};      Ruled Surface(109) = {108};
Line Loop(110) = {-11,-22,3,18};  Ruled Surface(111) = {110};
Line Loop(112) = {-13,-23,5,19};  Ruled Surface(113) = {112};
Line Loop(114) = {-6,19,14};      Ruled Surface(115) = {114};
Line Loop(116) = {-15,-24,7,20};  Ruled Surface(117) = {116};
Line Loop(118) = {-8,20,16};      Ruled Surface(119) = {118};
Line Loop(120) = {-31,-35,26,21}; Ruled Surface(121) = {120};
Line Loop(122) = {32,-22,-27,21}; Ruled Surface(123) = {122};
Line Loop(124) = {33,-23,-28,22}; Ruled Surface(125) = {124};
Line Loop(126) = {34,-24,-29,23}; Ruled Surface(127) = {126};

Surface Loop(128) = {93,-73,-55,95,-91};         Volume(129) = {128}; // int
Surface Loop(130) = {107,-75,-97,95,57,121};     Volume(131) = {130}; // int b
Surface Loop(132) = {105,-65,-97,-83,-93};       Volume(133) = {132}; // int h
Surface Loop(134) = {99,-111,77,123,59,107};     Volume(135) = {134}; // shell b
Surface Loop(136) = {99,-109,67,105,85};         Volume(137) = {136}; // shell h
Surface Loop(138) = {113,79,-101,-111,-125,-61}; Volume(139) = {138}; // ext b
Surface Loop(140) = {115,-69,-101,-87,-109};     Volume(141) = {140}; // ext h
Surface Loop(142) = {103,-117,-81,113,127,63};   Volume(143) = {142}; // inf b
Surface Loop(144) = {89,-119,71,103,115};        Volume(145) = {144}; // inf h

// Transfinite line commands explicitly specify the number of points
// and their distribution. 'Progression 2' means that each line
// element in the series will be twice as long as the preceding one.

Transfinite Line{35,21,22,23,24,38,17,18,19,20}   = nbpt_phi ;
Transfinite Line{31,26,48,44,42}                  = nbpt_int Using Progression 0.88;
Transfinite Line{41,37,36,9,11,1,3,13,5,15,7}     = nbpt_arc1 ;
Transfinite Line{30,25,40,39,10,2,12,4,14,6,16,8} = nbpt_arc2 ;
Transfinite Line{32,27,49,45,43}                  = nbpt_shell ;
Transfinite Line{33,28,46,50,52}                  = nbpt_far Using Progression 1.2 ;
Transfinite Line{34,29,51,47,53}                  = nbpt_inf Using Progression 1.05;

// *All* 2D and 3D transfinite entities are defined in respect to
// points. The ordering of the points defines the ordering of the mesh
// elements.

Transfinite Surface{55} = {1,14,16,18};
Transfinite Surface{57} = {14,2,19,16};
Transfinite Surface{59} = {2,3,21,19};
Transfinite Surface{61} = {3,4,23,21};
Transfinite Surface{63} = {4,5,25,23};
Transfinite Surface{73} = {1,15,17,18};
Transfinite Surface{75} = {15,10,20,17};
Transfinite Surface{77} = {10,11,22,20};
Transfinite Surface{79} = {11,12,24,22};
Transfinite Surface{81} = {12,13,26,24};
Transfinite Surface{65} = {18,16,19,6};
Transfinite Surface{67} = {6,19,21,7};
Transfinite Surface{69} = {7,21,23,8};
Transfinite Surface{71} = {8,23,25,9};
Transfinite Surface{83} = {17,18,6,20};
Transfinite Surface{85} = {20,6,7,22};
Transfinite Surface{87} = {22,7,8,24};
Transfinite Surface{89} = {24,8,9,26};
Transfinite Surface{91} = {1,14,15};
Transfinite Surface{95} = {15,14,16,17};
Transfinite Surface{93} = {18,16,17};
Transfinite Surface{121} = {15,14,2,10};
Transfinite Surface{97} = {17,16,19,20};
Transfinite Surface{123} = {10,2,3,11};
Transfinite Surface{99} = {20,19,21,22};
Transfinite Surface{107} = {10,2,19,20};
Transfinite Surface{105} = {6,20,19};
Transfinite Surface{109} = {7,22,21};
Transfinite Surface{111} = {11,3,21,22};
Transfinite Surface{101} = {22,21,23,24};
Transfinite Surface{125} = {11,3,4,12};
Transfinite Surface{115} = {8,24,23};
Transfinite Surface{113} = {24,12,4,23};
Transfinite Surface{127} = {12,13,5,4};
Transfinite Surface{103} = {24,23,25,26};
Transfinite Surface{119} = {9,26,25};
Transfinite Surface{117} = {13,5,25,26};

// As with Extruded meshes, the Recombine command tells Gmsh to
// recombine the simplices into quadrangles, prisms or hexahedra when
// possible. A colon in a list acts as in the 'For' loop: all surfaces
// having numbers between 55 and 127 are considered.

Recombine Surface {55:127};

// *All* 2D and 3D transfinite entities are defined in respect to
// points. The ordering of the points defines the ordering of the mesh
// elements.

Transfinite Volume{129} = {1,14,15,18,16,17};
Transfinite Volume{131} = {17,16,14,15,20,19,2,10};
Transfinite Volume{133} = {18,17,16,6,20,19};
Transfinite Volume{135} = {10,2,19,20,11,3,21,22};
Transfinite Volume{137} = {6,20,19,7,22,21};
Transfinite Volume{139} = {11,3,4,12,22,21,23,24};
Transfinite Volume{141} = {7,22,21,8,24,23};
Transfinite Volume{143} = {12,4,5,13,24,23,25,26};
Transfinite Volume{145} = {8,24,23,9,26,25};

VolInt           = 1000 ;
SurfIntPhi0      = 1001 ;
SurfIntPhi1      = 1002 ;
SurfIntZ0        = 1003 ;

VolShell         = 2000 ;
SurfShellInt     = 2001 ;
SurfShellExt     = 2002 ;
SurfShellPhi0    = 2003 ;
SurfShellPhi1    = 2004 ;
SurfShellZ0      = 2005 ;
LineShellIntPhi0 = 2006 ;
LineShellIntPhi1 = 2007 ;
LineShellIntZ0   = 2008 ;
PointShellInt    = 2009 ;

VolExt           = 3000 ;
VolInf           = 3001 ;
SurfInf          = 3002 ;
SurfExtInfPhi0   = 3003 ;
SurfExtInfPhi1   = 3004 ;
SurfExtInfZ0     = 3005 ;
SurfInfRight     = 3006 ;
SurfInfTop       = 3007 ;

Physical Volume  (VolInt)           = {129,131,133} ;
Physical Surface (SurfIntPhi0)      = {55,57,65} ;
Physical Surface (SurfIntPhi1)      = {73,75,83} ;
Physical Surface (SurfIntZ0)        = {91,121} ;

Physical Volume  (VolShell)         = {135,137} ;
Physical Surface (SurfShellInt)     = {105,107} ;
Physical Surface (SurfShellExt)     = {109,111} ;
Physical Surface (SurfShellPhi0)    = {59,67} ;
Physical Surface (SurfShellPhi1)    = {77,85} ;
Physical Surface (SurfShellZ0)      = {123} ;
Physical Line    (LineShellIntPhi0) = {1,2} ;
Physical Line    (LineShellIntPhi1) = {9,10} ;
Physical Line    (LineShellIntZ0)   = 21 ;
Physical Point   (PointShellInt)    = 6 ;

Physical Volume  (VolExt)           = {139,141} ;
Physical Volume  (VolInf)           = {143,145} ;
Physical Surface (SurfExtInfPhi0)   = {61,63,69,71} ;
Physical Surface (SurfExtInfPhi1)   = {79,87,81,89} ;
Physical Surface (SurfExtInfZ0)     = {125,127} ;
Physical Surface (SurfInfRight)     = {117} ;
Physical Surface (SurfInfTop)       = {119} ;

/********************************************************************* 
 *
 *  Gmsh tutorial 7
 * 
 *  Anisotropic meshes, Attractors
 *
 *********************************************************************/

// The new anisotropic 2D mesh generator can be selected with:

Mesh.Algorithm = 2 ;

// One can force a 4 step Laplacian smoothing of the mesh with:

Mesh.Smoothing = 4 ;

lc = .1;

Point(1) = {0.0,0.0,0,lc};
Point(2) = {1.2,-0.2,0,lc};
Point(3) = {1,1,0,lc};
Point(4) = {0,1,0,lc};

Line(1) = {3,2};
Line(2) = {2,1};
Line(3) = {1,4};
Line(4) = {4,3};

Line Loop(5) = {1,2,3,4};
Plane Surface(6) = {5};

Point(5) = {0.1,0.2,0,lc};
Point(11) = {0.5,0.5,-1,lc};
Point(12) = {0.5,0.5,0,lc};
Point(22) = {0.6,0.6,1,lc};

Line(5) = {11,22};

Spline(7) = {4,5,12,2};

// Anisotropic attractors can be defined on points and lines:

Attractor Line{5} = {.1, 0.01, 17};

Attractor Line{1,2} = {0.1, 0.005, 3};
Attractor Line{7} = {0.1, 0.05, 3};

/********************************************************************* 
 *
 *  Gmsh tutorial 8
 * 
 *  Post-Processing, Scripting, Animations, Options
 *
 *********************************************************************/

// The first example is included, as well as two post-processing maps
// (for the format of the post-processing maps, see the FORMATS file):

Include "t1.geo" ;
Include "view1.pos" ;
Include "view1.pos" ;

// Some general options are set (all the options specified
// interactively can be directly specified in the ascii input
// files. The current options can be saved into a file by selecting
// 'File->Save as->GEO complete options')...

General.Trackball = 0 ;
General.RotationX = 0 ;
General.RotationY = 0 ;
General.RotationZ = 0 ;
General.Color.Background = White ;
General.Color.Text = Black ;
General.Orthographic = 0 ;
General.Axes = 0 ;

// ...as well as some options for each post-processing view...

View[0].Name = "This is a very stupid demonstration..." ;
View[0].IntervalsType = 2 ;
View[0].OffsetZ = 0.05 ;
View[0].RaiseZ = 0 ;
View[0].Light = 1 ;

View[1].Name = "...of Gmsh's scripting capabilities" ;
View[1].IntervalsType = 1 ;
View[1].ColorTable = { Green, Blue } ;
View[1].NbIso = 10 ;

// ...and loop from 1 to 255 with a step of 1 is performed (to use a
// step different from 1, just add a third argument in the list,
// e.g. 'For num In {0.5:1.5:0.1}' increments num from 0.5 to 1.5 with
// a step of 0.1):

t = 0 ;

For num In {1:255}

  View[0].TimeStep = t ;
  View[1].TimeStep = t ;

  t = (View[0].TimeStep < View[0].NbTimeStep-1) ? t+1 : 0 ;
  
  View[0].RaiseZ += 0.01*t ;

  If (num == 3)
    // We want to use mpeg_encode to create a nice 320x240 animation
    // for all frames when num==3:
    General.GraphicsWidth = 320 ; 
    General.GraphicsHeight = 240 ;
  EndIf

  // It is possible to nest loops:

  For num2 In {1:50}

    General.RotationX += 10 ;
    General.RotationY = General.RotationX / 3 ;
    General.RotationZ += 0.1 ;
 
    Sleep 0.01; // sleep for 0.01 second
    Draw; // draw the scene

    If ((num == 3) && (num2 < 10))
      // The Sprintf function permits to create complex strings using
      // variables (since all Gmsh variables are treated internally as
      // double precision numbers, the format should only contain valid
      // double precision number format specifiers):
      Print Sprintf("t8-0%g.jpg", num2);
    EndIf

    If ((num == 3) && (num2 >= 10))
      Print Sprintf("t8-%g.jpg", num2);
    EndIf

  EndFor

  If(num == 3)
    // We make a system call to generate the mpeg
    System "mpeg_encode t8.par" ;    
  EndIf

EndFor


// Here is the list of available scripting commands:
//  
//  Merge string;     (to merge a file)
//  Draw;             (to draw the scene)
//  Mesh int;         (to perform the mesh generation; 'int' = 0, 1, 2 or 3)
//  Save string;      (to save the mesh)
//  Print string;     (to print the graphic window)
//  Sleep expr;       (to sleep during expr seconds)
//  Delete View[int]; (to free the view int)
//  Delete Meshes;    (to free all meshes)
//  System string;    (to execute a system call)

/********************************************************************* 
 *
 *  Gmsh tutorial 9
 * 
 *  Post-Processing, Plugins
 *
 *********************************************************************/

// Plugins can be added to Gmsh in order to extend its
// capabilities. For example, post-processing plugins can modify a
// view, or create a new view based on previously loaded
// views. Several default plugins are statically linked into Gmsh,
// e.g. CutMap, CutPlane, CutSphere, Skin, Transform or Smooth.

// Let's load a three-dimensional scalar view

Include "view3.pos" ;

// Plugins can be controlled in the same way as other options in
// Gmsh. For example, the CutMap plugin (which extracts an isovalue
// surface from a 3D scalar view) can either be called from the
// graphical interface (right click on the view button, then
// Plugins->CutMap), or from the command file:

Plugin(CutMap).A = 0.67 ; // iso-value level
Plugin(CutMap).iView = 0 ; // source view is View[0]
Plugin(CutMap).Run ; 

// The following runs the CutPlane plugin:

Plugin(CutPlane).A = 0 ; 
Plugin(CutPlane).B = 0.2 ; 
Plugin(CutPlane).C = 1 ; 
Plugin(CutPlane).D = 0 ; 
Plugin(CutPlane).Run ; 

View[0].Light = 1;
View[0].IntervalsType = 2;
View[0].NbIso = 6;
View[0].SmoothNormals = 1;

View[1].IntervalsType = 2;

View[2].IntervalsType = 2;
Draw;