Lab 03 - Poisson Problem

Theory and Practice of Finite Elements

Luca Heltai [email protected]

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General Instructions

For each of the point below, extend the step-3 class with functions that perform the indicated tasks, trying to minimize the amount of code you copy and paste, possibly restructuring existing code by adding arguments to existing functions, and generating wrappers similar to the run method (e.g., run_exercise_3).

Once you created a function that performs the given task, add it to the step-3-tester.cc file, and make sure all the exercises are run through the gtest executable, e.g., adding a test for each exercise, as in the following snippet:

TEST_F(Step3Tester, Exercise3) {
   run_exercise_3();
}

By the end of this laboratory, you will have a code that solves a Poisson problem in two dimensions, on different domain types, with arbitrary right hand side, arbitrary Dirichlet boundary conditions, variable finite element degree, and different levels of refinement.

The program will read parameter files for its execution. This is the minimal starting point for simple but extensible Finite Element applications, and touches already many important and fundamental characteristics of general and extensible Finite Element codes.

Lab-03

step-3

1. See documentation of step-3 at https://www.dealii.org/current/doxygen/deal.II/step_3.html

2. Compile and run step-3. Examine the source and header files. Move all #include<...> directives that you can from the header step-3.h to the source file step-3.cc.

3. Open the vtk output and visualize the solution in paraview or visit. Figure out how to warp the solution by the solution variable. Save a picture of your visualization on the figures subdirectory, named step-3-0.png, and commit it to your repository.

4. Follow the instructions in “Modify the type of boundary condition” in the description of the tutorial, plot your output, and save a figure in the figures subdirectory, named step-3-1.png, and commit it to your repository. Make sure the test is run through gtest.

5. Now also do “A slight variation of the last point” but use the value -0.5 for the boundary with indicator 1. Save your output as step-3-2.png, and commit it to the figures directory in your repository. Make sure the test is run through gtest.

6. Change the setup to have $f=0$. Save your output as step-3-3.png, and commit it to the figures directory in your repository. Make sure the test is run through gtest.

7. Switch to an L-shaped domain and experiment with a combination of Dirichlet and Neumann boundary conditions. By experimentation, identify the faces adjacent to the re-entrant corner and apply Dirichlet conditions only there. Save your output as step-3-4.png, and commit it to the figures directory in your repository. Make sure the test is run through gtest.

8. Bonus: Do “Convergence of the mean” (read through the end of the documentation of ). Can you see the order $h^2$? Increase the polynomial order (you need to increase all orders of the quadratures in the program!) and check the convergence of the mean now. Make sure the test is run through gtest.

ParameterHandler

1. Follow the documentation of the class ParameterAcceptor (from "A typical usage of this class"). Derive your Step3 class from ParameterAcceptor, and initialize the ParameterAcceptor class using the string "Step3". Add a function that initializes the class given a parameter file name, i.e., initialize("parameters.prm"), by calling ParameterAcceptor::initialize(...)

2. Add the following parameters to your Step3 class, and make sure they are used throughout your code

   • Finite element degree
   • Number of global refinements
   • Output filename

   Notice that the FE_Q class will need to be built after the initialization of the Finite element degree variable. You can create it right before you use it, and then reference to it by DoFHandler::get_fe() when you need it later.

3. Add two FunctionParser<2> members to your Step3 class, one for the boundary conditions, and one for the forcing term. Add two std::string members, representing their mathematical expressions, and a std::map<std::string, double> representing the constants to use in your mathematical expression, and make sure they are added as parameters of your class as

   • Forcing term expression
   • Boundary condition expression
   • Problem constants

   and initialize the two FunctionParser<2> objects with the given expressions and the given constants. Make sure your Step3 class uses these two functions where the boundary conditions are computed and where the right hand side is assembled

4. Following the ideas in step-70, add two additional parameters to the Step3 class to select at run time what grid to create, and with what arguments. Use as parameters

   • Grid generator function
   • Grid generator arguments

5. Test your application with all the files in the parameters subdirectory, (running the test through the gtest application), and create a visualization for each of the given parameters, with the same name of the parameter file, i.e., the visualization of the test that runs parameters/hyper_shell.prm should be saved on an image named figures/hyper_shell.png. Commits all your generated figures. Notice that for older versions of google test, you may need to run each test individually (this is a known issue between google testsuite and the ParameterAcceptor class). To run an individual test, you can use the --gtest_filter command line option, i.e., the following command

   ./gtest --gtest_filter=Step3Tester.HyperShell
   

   will run only the test Step3Tester.HyperShell.

   If you want to run all tests one after the other you can call the ctest utility, which will execute the gtest command passing the gtest filter for each of the tests you provided, producing, for example, an output similar to:

   dealii@66c3be678611:/workspace/sissa-mhpc-lab-03/build-container$ ctest
   Test project /workspace/sissa-mhpc-lab-03/build-container
       Start 1: Step3Tester.MakeGrid
   1/6 Test #1: Step3Tester.MakeGrid .............   Passed    0.68 sec
       Start 2: Step3Tester.HyperCube
   2/6 Test #2: Step3Tester.HyperCube ............   Passed    0.78 sec
       Start 3: Step3Tester.HyperShell
   3/6 Test #3: Step3Tester.HyperShell ...........   Passed    1.90 sec
       Start 4: Step3Tester.HyperBall
   4/6 Test #4: Step3Tester.HyperBall ............   Passed    1.28 sec
       Start 5: Step3Tester.SinRhs
   5/6 Test #5: Step3Tester.SinRhs ...............   Passed    0.79 sec
       Start 6: Step3Tester.SinBc
   6/6 Test #6: Step3Tester.SinBc ................   Passed    1.26 sec
   
   100% tests passed, 0 tests failed out of 6