get hexahedral meshes. 3D meshing of unstructured geometry into hexahedra is not
supported at the time of writing this tutorial (early 2013).
-In gmsh, a mesh is described in a text based .geo file, that can be edited by
-hand and
-can contain computations, loops, variables, etc. It is very flexible. The
-mesh is generated from a surface representation, which is build from a list of
-line loops, which is build from a list of lines, which are in turn built from
-points.
-
-It is important that file contains "physical lines" and "physical
-surfaces". These give the boundary indicators and material ids for the use
-inside deal.II. Without the physical entities, nothing will be imported into
-deal.II. The .geo script can be written by hand or it can be generated
-automatically by creating objects graphically. In many cases it is best to
-combine both approaches. The file can be easily reloaded by pressing "reload"
-under the "Geometry" tab.
-
-This tutorial contains an example .geo file, that describes a box with two
-objects cut out in the interior. This is how untitled.geo looks like in gmsh
-(displaying the boundary indicators):
-@image html gmsh_picture.png Mesh displayed in Gmsh
+In gmsh, a mesh is described in a text based <code>.geo</code> file, that can
+contain computations, loops, variables, etc. It is very flexible. The mesh is
+generated from a surface representation, which is build from a list of line
+loops, which is build from a list of lines, which are in turn built from
+points. The <code>.geo</code> script can be written and edited by hand or it
+can be generated automatically by creating objects graphically inside gmsh. In
+many cases it is best to combine both approaches. The file can be easily
+reloaded by pressing "reload" under the "Geometry" tab.
+
+This tutorial contains an example <code>.geo</code> file, that describes a box
+with two objects cut out in the interior. This is how
+<code>untitled.geo</code> looks like in gmsh (displaying the boundary
+indicators as well as the mesh discussed further down below):
-You might want to open the .geo file in a text editor to see how it is
-structured. Deal.II can read the .msh format written by gmsh. You generate the
-.msh from the .geo by running:
+@image html gmsh_picture.png Mesh displayed in Gmsh
-<pre>
+You might want to open the <code>untitled.geo</code> file in a text editor (it
+is located in the same directory as the <code>step-49.cc</code> source file) to
+see how it is structured. You can see how the boundary of the domain is
+composed of a number of lines and how later on we combine several lines into
+"physical lines" (or "physical surfaces") that list the logical lines'
+numbers. "Physical" object are the ones that carry information about the
+boundary indicator (see @ref GlossBoundaryIndicator "this glossary entry").
+
+@note It is important that this file contain "physical lines" and "physical
+surfaces". These give the boundary indicators and material ids for use
+in deal.II. Without these physical entities, nothing will be imported into
+deal.II.
+
+deal.II's GridIn class can read the <code>.msh</code> format written by
+gmsh and that contains a mesh created for the geometry described by the
+<code>.geo</code> file. You generate the <code>.msh</code> from the
+<code>.geo</code> by running the commands
+
+@code
gmsh -2 untitled.geo
-</pre>
+@endcode
-or by clicking "Mesh" and then "2D" inside Gmsh after loading the file. Now
-this is the mesh read from the .msh file and saved again by deal.II as an
-image (see the grid_1 function):
+on the command line, or by clicking "Mesh" and then "2D" inside Gmsh after
+loading the file. Now this is the mesh read from the <code>.msh</code> file
+and saved again by deal.II as an image (see the <code>grid_1</code> function
+of the current program):
<img src="http://www.dealii.org/images/steps/developer/step-49.grid-1.png" alt="">
those functions here.
The function GridTools::transform allows you to transform the vertices of a
-given mesh using a smooth function.
-
-In the function grid_5() we perturb the y coordinate of a mesh with
-a sin curve:
+given mesh using a smooth function. An example of its use is also given in the
+results section of step-38 but let us show a simpler example here:
+In the function <code>grid_5()</code> of the current program, we perturb the y
+coordinate of a mesh with a sine curve:
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
</tr>
</TABLE>
-Using the formula
-$(x,y) \mapsto (x,tanh(2*y)/tanh(2))$, we transform a regular refined
-unit square to a wall-adapted mesh in y direction. This is done in grid_6()
+Similarly, we can transform a regularly refined
+unit square to a wall-adapted mesh in y direction using the formula
+$(x,y) \mapsto (x,\tanh(2*y)/\tanh(2))$. This is done in <code>grid_6()</code>
of this tutorial:
<TABLE WIDTH="60%" ALIGN="center">
<tr>
</tr>
</TABLE>
-The function Triangulation::distort_random allows you to move vertices in the
+Finally, the function Triangulation::distort_random allows you to move vertices in the
mesh (optionally ignoring boundary nodes) by a random amount. This is
-demonstrated in grid_7() and the result is as follows:
+demonstrated in <code>grid_7()</code> and the result is as follows:
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
</tr>
</TABLE>
-Please note that while this allows you to negate some of the superconvergence
-effects one gets when studying convergence on regular meshes, it is better to
-work with a sequence of unstructured meshes (see possible extensions).
+This function is primarily intended to negate some of the superconvergence
+effects one gets when studying convergence on regular meshes, as well as to
+suppress some optimizations in deal.II that can exploit the fact that cells
+are similar in shape. In practice, it is of course always better to
+work with a sequence of unstructured meshes (see possible extensions at the
+end of the this section).
<h4>Merging Meshes</h4>
The function GridGenerator::merge_triangulations() allows you to merge two
-given Triangulations into a single one. For this to work, the vertices of the
-shared edge or face have to match exactly. Lining up the two meshes can be
-achieved using GridTools::shift and GridTools::scale. In the function
-grid_2() of this tutorial, we merge a square with a round hole (generated with
-hyper_cube_with_cylindrical_hole()) and a rectangle. The function
-GridTools::subdivided_hyper_rectangle() allows you to specify the number of
-repetitions and the positions of the corners, this way we do not need to shift
-it manually. You should inspect the mesh graphically to make sure, that cells
-line up correctly and no hanging nodes exist in the merged Triangulation.
+given Triangulation objects into a single one. For this to work, the vertices
+of the shared edge or face have to match exactly. Lining up the two meshes
+can be achieved using GridTools::shift and GridTools::scale. In the function
+<code>grid_2()</code> of this tutorial, we merge a square with a round hole
+(generated with GridGenerator::hyper_cube_with_cylindrical_hole()) and a
+rectangle (generated with GridGenerator::subdivided_hyper_rectangle()). The
+function GridGenerator::subdivided_hyper_rectangle() allows you to specify the
+number of repetitions and the positions of the corners, so there is no need to
+shift the triangulation manually here. You should inspect the mesh graphically
+to make sure that cells line up correctly and no unpaired nodes exist in the
+merged Triangulation.
These are the input meshes and the output mesh:
<TABLE WIDTH="80%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-2a.png" alt=""> input mesh 1
+ <img
+ src="http://www.dealii.org/images/steps/developer/step-49.grid-2a.png"
+ alt="" height="200px"> input mesh 1
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-2b.png" alt=""> input mesh 2
+ <img src="http://www.dealii.org/images/steps/developer/step-49.grid-2b.png" alt="" height="200px"> input mesh 2
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-2.png" alt=""> merged mesh
+ <img src="http://www.dealii.org/images/steps/developer/step-49.grid-2.png" alt="" height="200px"> merged mesh
</td>
</tr>
</table>
<h4>Moving Vertices</h4>
-The function grid_3() demonstrates the ability to pick individual vertices and
-move them around in an existing mesh. Please be sure to not produce degenerate
-cells. Here, we create a box with a cylindrical hole, that is not exactly
+The function <code>grid_3()</code> demonstrates the ability to pick individual vertices and
+move them around in an existing mesh. Note that this has the potential to produce degenerate
+or inverted cells and you shouldn't expect anything useful to come of using
+such meshes. Here, we create a box with a cylindrical hole that is not exactly
centered by moving the top vertices upwards:
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-3a.png" alt=""> input mesh
+ <img src="http://www.dealii.org/images/steps/developer/step-49.grid-3a.png" alt="" height="200px"> input mesh
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-3.png" alt=""> top vertices moved upwards
+ <img src="http://www.dealii.org/images/steps/developer/step-49.grid-3.png" alt="" height="200px"> top vertices moved upwards
</td>
</tr>
</table>
+For the exact way how this is done, see the code below.
+
<h4>Extruding Meshes</h4>
If you need a 3d mesh that can be created by extruding a given 2d mesh (that
can be created in any of the ways given above), you can use the function
-GridGenerator::extrude_triangulation(). See grid_4() function in this tutorial
-for an example. Note that the given result could also be achieved using the 3d
-version of hyper_cube_with_cylindrical_hole(). The main usage is a 2d mesh
-generated for example with gmsh, that is read in from a .msh file as described
-above. This is the output from grid_4():
+GridGenerator::extrude_triangulation(). See the <code>grid_4()</code> function
+in this tutorial for an example. Note that for this particular case, the given
+result could also be achieved using the 3d version of
+GridGenerator::hyper_cube_with_cylindrical_hole(). The main usage is a 2d
+mesh, generated for example with gmsh, that is read in from a
+<code>.msh</code> file as described above. This is the output from grid_4():
<TABLE WIDTH="60%" ALIGN="center">
<tr>
</table>
+<!--
+
<h4>Possible Extensions</h4>
- Modify a mesh:
- Database of unstructured meshes for convergence studies
- discuss GridTools::extract_boundary_mesh
- how to remove or disable a cell inside a mesh
+-->
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