README.md

deal.II Crash Course of the 10th deal.II Users and Developers Workshop

This repository contains the source code for the deal.II crash course of the 10th deal.II Users and Developers Workshop. In addition, the accompanying presentation can be found here.

The course is structured in two sessions. In the first session, the basic layout of a deal.II program will be covered by solving a stationary Poisson's problem. The second session will be dedicated to the solution of a time-dependent problem. More concrete, a linear Heat equation.

In both case, a code skeleton is provided which should be filled in a gap text manner. The sessions' folders can be found by their names. In addition, each session folder contains a solution folder which includes a working code solution.

Run Code

1. cmake -DDEAL_II_DIR=PATH2DEALII .: build
2. make release: switch from debug to release mode
3. make: compile code
4. make run: run code

Session 1

Session 1 aims to solve Poisson's problem and is oriented on step-3 of the deal.II tutorial program.

Poisson's problem

Let $\Omega\subset\mathbb{R}^d$ and dimension $d=1,2,3$. Find $u:\bar{\Omega}\to\mathbb{R}$ such that

$$ \begin{aligned} -\Delta u &= f &&\quad\text{in } \Omega\\ u &= g &&\quad\text{on } \partial\Omega \end{aligned} $$

In this exercise, let:

• $\Omega = (-1,1)^2$
• $f = 1$,
• $g = 0$.

Exemplary solutions for $d=1,2$ are sketched or shown below.

Exercise

The missing gaps are marked by /* MISSING CODE /* comments. The following steps must be done to complete the code:

1. Complete Poisson::run().
   • This method is used as a central place to call the other methods of the class to solve the problem.
2. Complete Poisson::make_grid() by generating the mesh, c.f. deal.II GridGenerator.
3. Complete Poisson::assemble_system().
   1. Get the FEValues<2> fe_values, c.f. FEValues Constructor and FEValues update flags.
   2. Reset the local cell's contributions.
   3. Assemble system matrix contributions.
   4. Assemble the rhs contribution.
   5. Transfer the local elements to the global matrix.
   6. Transfer the local elements to the global rhs.
   7. Set Dirichlet boundary conditions.

Bonus

For further Interaction with the code, there are two bonus exercises that could be done to extend the code. These are marked in the code by /* BONUS */.

1. Try out and replace the homogeneous Dirichlet BC by non-zero BC.
2. Investigate the solution on successively refined meshes.

Session 2

Heat equation

Let $\Omega\subset \mathbb{R}^d$ as before and $I=(0,T)$, with $T>0$. Find $u:=u(x,t):\bar{\Omega}\times \bar{I}\to \mathbb{R}$ such that

$$ \begin{aligned} \rho \partial_t u - \nabla\cdot (\alpha(x) \nabla u) &= f &&\quad\text{in } \Omega\times I,\\ u &= u_D &&\quad\text{on } \partial\Omega \times (0,T),\\ u(0) &= u_0 &&\quad\text{in } \Omega\times {t=0}, \end{aligned} $$

where $f:\Omega\times I\to \mathbb{R}$ and $u_0:\Omega\to\mathbb{R}$ and $\alpha(x)>0$ and $\rho>0$ are material parameters, and $u_D\geq 0$ is a Dirichlet boundary condition. As an example, $u_0$ is the initial temperature and $u_D$ is the wall temperature, and $f$ is some heat source.

In this exercise, let $\Omega = (0,\pi)^2$ and $I = (0,20)$ and

• $u_D = 0$,
• $u_0 = \sin(x_1)\sin(x_2)$,
• $f = 0$,
• $\alpha(x) = 1$.

The solution for $u$ is illustrated below.

Exercise

The missing gaps are marked by /* MISSING CODE /* comments. The following steps must be done to complete the code:

1. In main(): Initialize a 2D instance of the Step_Heat class and start the solution process.
2. In Step_Heat<dim>::run(): Call the grid generation and the system setup.
3. In Step_Heat<dim>::setup_system():
   1. Create the sparsity pattern.
   2. Initialize the rhs and solutions vectors.
      • Note, that a vector to store the old solution is required.
4. In InitialValues<dim>::value: Add the initial condition for the solution.
5. In Step_Heat<dim>::run():
   1. Call the assemble of the system for each new time step solve.
   2. Solve the LES.
   3. Add the stopping criterion for the time step loop.
6. In Step_Heat<dim>::assemble_system():
   1. Add the fe_values.
   2. Add mass matrix contribution to cell_matrix.
   3. Add stiffness matrix contribution to cell_matrix.
   4. Add rhs values contributions to cell_vector.
   5. Add old time step solution contributions to cell_vector.

Bonus

For further Interaction with the code, there are two bonus exercises that could be done to extend the code. These are marked in the code by /* BONUS 1 */ and /* BONUS 2 */, respectively.

1. The code contains the possibility to choose a space-dependent $\alpha(x) \leq 0$. Try to simulate the heat equation with two different heat coefficients.
   1. In Coefficient<dim>::value: Return a space dependent heat coefficient.
   2. In assemble_system: Add the const Coefficient<dim> coefficient and its contributions in the assembly of cell_matrix.
2. Previously, we choose a homogenous RHS. Try to extend this top a non-zero RHS.

Contact

Should you have any questions do not hesitate to send us an email at

fischer@ifam.uni-hannover.de
wick@ifam.uni-hannover.de