Lab 02 - Triangulation, DoFHandler, FiniteElement

Theory and Practice of Finite Elements

Luca Heltai [email protected]

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1. Setup

   • You will need the following programs:
     • Visual Studio Code with the Remote Development plugin installed
     • Paraview (ver > 5.4 would be better)
   • Check that the .devcontainer directory contains a configuration that works for you. When you open the root directory of the repository, it should prompt you to reopen in container. If not, try explictily with the command F1 -> Reopen folder in container
   • If opening in container worked, you should be able to run a terminal (CTRL + ~) and see something like

   To run a command as administrator (user "root"), use "sudo <command>".
   See "man sudo_root" for details.
   
   dealii@3f25621373bd:/workspaces/02_tria_dh_fem$ 
   

   • The folder contains a CMakeLists.txt file. VSCode should detect this, and provide commands to configure and build your application, at the bottom of the terminal window

2. Tasks for lab-02:

   1. Read the documentation of step-1 deal.II at\

      • https://www.dealii.org/current/doxygen/deal.II/step_1.html
      • https://www.dealii.org/current/doxygen/deal.II/step_2.html

   2. Compile and run step-1 inside the VSCode container and look at the output.

   3. Comment out the .set_manifold(0, ...) line in second_grid(). What happens now?

   4. Create an image of an L-shape domain (add a function third_grid() to step-1) with one global refinement.

   5. Now change the output format of the previous example to vtk and open the new file in paraview.

   6. Refine the L-shaped mesh adaptively around the re-entrant corner several times but with a twist: refine all cells with the distance between the center of the cell and re-entrant corner is smaller than 1/3.

   7. Output mesh two as an svg file instead of eps. Open it in a browser to display it (firefox for example).

   8. Create a helper function that takes a reference to a Triangulation and returns a tuple with the following information: number of levels, number of cells, number of active cells. Test this with all of your meshes.

   9. Generate a circle using GridGenerator::hyper_ball() in 2d: use a SphericalManifold everywhere, only on the boundary, or on all cells except the center cell and refine the mesh globally twice.

   10. Go into second_grid() and add a triangulation.reset_manifold(0) right before the refinment loop. Explain in a comment of the code what is happening.

   11. Bonus: Create a mesh that represents the surface of a torus and refine it 2 times globally. Output to vtk format and check the output. Note that your Triangulation needs to be of type Triangulation<2,3>, which we will discuss later this week.

   12. Bonus: Take a look at step-49 and read the included .msh file in your modified step-1 program.

   13. Read the documentation of step-2 deal.II at\

   • https://www.dealii.org/current/doxygen/deal.II/step_2.html

   14. Run step-2. Look at the sparsity patterns in firefox.

   15. How does the pattern change if you increase the polynomial degree from 1 to 2 or to 3?

   16. How does the pattern change if you use a globally refined (say 3 times) unit square?

   17. Are these patterns symmetric? Why/why not?

   18. How many entries per row in the sparsity pattern do you expect for a Q1 element (assuming four cells are around each vertex)? Check that this is true for the mesh in b) (look for row_length(i) and output them for each row). Can you construct a 2d mesh (without hanging nodes) that has a row with more entries?

   19. How many entries per row in the sparsity pattern are there for Q2 and Q3 elements, again assuming four cells around each vertex?

   20. Print all entries for row 42 for the original renumbered sparsity pattern.

   21. Bonus: Compute and output statistics like the number of unknowns, bandwidth of the sparsity pattern, average number of entries per row, and fill ratio.

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.svg output files with grids and sparsity patterns are in folder images