From 0bbdf76b411efaa87240b0d0c7a0a2feacc3f11a Mon Sep 17 00:00:00 2001
From: Mathis Kelm <mathis.kelm@iws.uni-stuttgart.de>
Date: Tue, 7 Mar 2023 15:52:20 +0100
Subject: [PATCH] [examples] Add description for cahn-hilliard model

---
 examples/README.md                            |  12 +
 examples/cahn_hilliard/.doc_config            |  27 +
 examples/cahn_hilliard/README.md              |  70 +++
 examples/cahn_hilliard/doc/_intro.md          |  57 ++
 examples/cahn_hilliard/doc/mainfile.md        | 492 ++++++++++++++++++
 examples/cahn_hilliard/doc/mainfile_intro.md  |   0
 examples/cahn_hilliard/doc/modelheader.md     | 394 ++++++++++++++
 .../cahn_hilliard/doc/modelheader_intro.md    |   0
 examples/cahn_hilliard/img/animation.gif      | Bin 0 -> 2687539 bytes
 examples/cahn_hilliard/main.cc                | 282 +++++++---
 examples/cahn_hilliard/model.hh               | 176 +++++--
 11 files changed, 1411 insertions(+), 99 deletions(-)
 create mode 100644 examples/cahn_hilliard/.doc_config
 create mode 100644 examples/cahn_hilliard/README.md
 create mode 100644 examples/cahn_hilliard/doc/_intro.md
 create mode 100644 examples/cahn_hilliard/doc/mainfile.md
 create mode 100644 examples/cahn_hilliard/doc/mainfile_intro.md
 create mode 100644 examples/cahn_hilliard/doc/modelheader.md
 create mode 100644 examples/cahn_hilliard/doc/modelheader_intro.md
 create mode 100644 examples/cahn_hilliard/img/animation.gif

diff --git a/examples/README.md b/examples/README.md
index beb935118d..057987caa5 100644
--- a/examples/README.md
+++ b/examples/README.md
@@ -224,3 +224,15 @@ __Discretization method:__ Cell-centered finite volumes with two-point flux appr
 <figure><img src="embedded_network_1d3d/img/network.png" alt="blood vessel network"/></figure></td>
 </a></td>
 </tr></table>
+
+### [:open_file_folder: Example 11: The Cahn-Hilliard model, a nonlinear PDE](cahn_hilliard/README.md)
+
+<table><tr><td>
+
+In this example we simulate the separation of two immiscible phases using the Cahn-Hilliard model.
+
+</td>
+<td width="35%"><a href="cahn_hilliard/README.md">
+<figure><img src="cahn_hilliard/img/animation.gif" alt="phase distribution"/></figure></td>
+</a></td>
+</tr></table>
diff --git a/examples/cahn_hilliard/.doc_config b/examples/cahn_hilliard/.doc_config
new file mode 100644
index 0000000000..d1a87cc8a3
--- /dev/null
+++ b/examples/cahn_hilliard/.doc_config
@@ -0,0 +1,27 @@
+{
+    "README.md" : [
+        "doc/_intro.md"
+    ],
+
+    "doc/mainfile.md" : [
+        "doc/mainfile_intro.md",
+        "main.cc"
+    ],
+
+    "doc/modelheader.md" : [
+        "doc/modelheader_intro.md",
+        "model.hh"
+    ],
+
+    "navigation" : {
+        "mainpage" : "README.md",
+        "subpages" : [
+            "doc/modelheader.md",
+            "doc/mainfile.md"
+        ],
+        "subtitles" : [
+            "Model implementation",
+            "Main program"
+        ]
+    }
+}
diff --git a/examples/cahn_hilliard/README.md b/examples/cahn_hilliard/README.md
new file mode 100644
index 0000000000..a192a3346b
--- /dev/null
+++ b/examples/cahn_hilliard/README.md
@@ -0,0 +1,70 @@
+<!-- Important: This file has been automatically generated by generate_example_docs.py. Do not edit this file directly! -->
+
+# Cahn-Hilliard model
+
+A random initial distribution of two phases separating according to the Cahn-Hilliard model.
+
+__The main points illustrated in this example are__
+* A base setup for solving a nonlinear partial differential equation
+
+__Table of contents__. This description is structured as follows:
+
+[[_TOC_]]
+
+__Result__. The result will look like below.
+You see the concentration variable $c$ capturing the phase distribution after 1 second.
+
+<figure>
+    <center>
+        <img src="img/results_phase_distribution.png" alt="Concentration field modeling phase distribution" width="50%"/>
+    </center>
+</figure>
+
+## Problem set-up
+
+A square two-dimensional domain is initialized with a concentration `c`, randomly perturbed by
+noise following a uniform distribution between 0.41 and 0.43. 
+With time the concentration field evolves towards attaining mostly values near to 0 or 1 while
+conserving the total concentration, modeling the separation of two immiscible fluids.
+
+## Model description
+
+The Cahn-Hilliard model uses a pair of second order nonlinear partial differential equations,
+for a concentration field $c$ and a chemical potential $\mu$. The former is conserved and
+evolves with a flux proportional to $\nabla \mu$, while the latter depends on the laplacian of
+the concentration $\nabla^2 c$ and a nonlinear function of $c$, a derivative of a free energy
+functional.
+
+```math
+\frac{\partial c}{\partial t} = M \nabla^2 \mu \\
+\mu = E (4 c^3 - 6 c^2 +2 c) - \gamma \nabla^2 c
+```
+
+Here $M$ denotes the mobility of the concentration, while the energy scale $E$ and surface tension
+$\gamma$ balance the two contributions of the free energy functional.
+
+## Implementation of a simple nonlinear PDE
+
+For this example the C++ code is contained in two files, `main.cc` and `model.hh`.
+The `model.hh` header contains the `ModelTraits` and properties for the model TypeTag
+`CahnHilliardModel`, as well as volumevariables `CahnHilliardModelVolumeVariables`
+and the basic local residual `CahnHilliardModelLocalResidual`.
+The residual's storage term is extended in the problem definition `CahnHilliardTestProblem`
+in the file `main.cc`, which also contains more specific properties of the problem setup (for
+type tag `CahnHilliardTest`) as well as the actual simulation loop usually found in a mainfile.
+
+More details are given in [main.cc](doc/mainfile.md) and [model.hh](doc/modelheader.md).
+
+Additionally the folder contains `params.input` containing runtime parameters and the build
+system file `CMakeLists.txt` defining conditions for the accompanying test.
+
+## Part 1: Model implementation
+
+| [:arrow_right: Click to continue with part 1 of the documentation](doc/modelheader.md) |
+|---:|
+
+
+## Part 2: Main program
+
+| [:arrow_right: Click to continue with part 2 of the documentation](doc/mainfile.md) |
+|---:|
diff --git a/examples/cahn_hilliard/doc/_intro.md b/examples/cahn_hilliard/doc/_intro.md
new file mode 100644
index 0000000000..0c63dab293
--- /dev/null
+++ b/examples/cahn_hilliard/doc/_intro.md
@@ -0,0 +1,57 @@
+# Cahn-Hilliard model
+
+A random initial distribution of two phases separating according to the Cahn-Hilliard model.
+
+__The main points illustrated in this example are__
+* A base setup for solving a nonlinear partial differential equation
+
+__Table of contents__. This description is structured as follows:
+
+[[_TOC_]]
+
+__Result__. The result will look like below.
+You see the concentration variable $c$ capturing the phase distribution after 1 second.
+
+<figure>
+    <center>
+        <img src="img/results_phase_distribution.png" alt="Concentration field modeling phase distribution" width="50%"/>
+    </center>
+</figure>
+
+## Problem set-up
+
+A square two-dimensional domain is initialized with a concentration `c`, randomly perturbed by
+noise following a uniform distribution between 0.41 and 0.43. 
+With time the concentration field evolves towards attaining mostly values near to 0 or 1 while
+conserving the total concentration, modeling the separation of two immiscible fluids.
+
+## Model description
+
+The Cahn-Hilliard model uses a pair of second order nonlinear partial differential equations,
+for a concentration field $c$ and a chemical potential $\mu$. The former is conserved and
+evolves with a flux proportional to $\nabla \mu$, while the latter depends on the laplacian of
+the concentration $\nabla^2 c$ and a nonlinear function of $c$, a derivative of a free energy
+functional.
+
+```math
+\frac{\partial c}{\partial t} = M \nabla^2 \mu \\
+\mu = E (4 c^3 - 6 c^2 +2 c) - \gamma \nabla^2 c
+```
+
+Here $M$ denotes the mobility of the concentration, while the energy scale $E$ and surface tension
+$\gamma$ balance the two contributions of the free energy functional.
+
+## Implementation of a simple nonlinear PDE
+
+For this example the C++ code is contained in two files, `main.cc` and `model.hh`.
+The `model.hh` header contains the `ModelTraits` and properties for the model TypeTag
+`CahnHilliardModel`, as well as volumevariables `CahnHilliardModelVolumeVariables`
+and the basic local residual `CahnHilliardModelLocalResidual`.
+The residual's storage term is extended in the problem definition `CahnHilliardTestProblem`
+in the file `main.cc`, which also contains more specific properties of the problem setup (for
+type tag `CahnHilliardTest`) as well as the actual simulation loop usually found in a mainfile.
+
+More details are given in [main.cc](doc/mainfile.md) and [model.hh](doc/modelheader.md).
+
+Additionally the folder contains `params.input` containing runtime parameters and the build
+system file `CMakeLists.txt` defining conditions for the accompanying test.
diff --git a/examples/cahn_hilliard/doc/mainfile.md b/examples/cahn_hilliard/doc/mainfile.md
new file mode 100644
index 0000000000..ef55ebc606
--- /dev/null
+++ b/examples/cahn_hilliard/doc/mainfile.md
@@ -0,0 +1,492 @@
+<!-- Important: This file has been automatically generated by generate_example_docs.py. Do not edit this file directly! -->
+
+
+| [:arrow_left: Back to the main documentation](../README.md) | [:arrow_left: Go back to part 1](modelheader.md) |
+|---|---:|
+
+
+
+# Problem, test properties/traits and main program flow (`main.cc`)
+In this example the file `main.cc` contains the problem class `CahnHilliardTestProblem`,
+properties and traits specific to the test case as well as the main program flow in the form of
+`main` function.
+
+
+```cpp
+#include <config.h>
+```
+
+## Problem
+The __problem class__ defines boundary conditions and extends the storage term defined in the
+model's localresidual by the derivative of the free energy.
+
+
+<details open>
+<summary><b>Click to hide/show the file documentation</b> (or inspect the [source code](../main.cc))</summary>
+
+### Include headers
+
+
+```cpp
+// use the property system and runtime parameters
+#include <dumux/common/properties.hh>
+#include <dumux/common/parameters.hh>
+// common DuMux vector for discretized equations
+#include <dumux/common/numeqvector.hh>
+// types of boundary conditions to use
+#include <dumux/common/boundarytypes.hh>
+// generic problem for finite volume simulations
+#include <dumux/common/fvproblem.hh>
+```
+
+
+### The problem class `CahnHilliardTestProblem`
+In this abstract problem we inherit from the generic `FVProblem`.
+
+
+```cpp
+namespace Dumux {
+
+template<class TypeTag>
+class CahnHilliardTestProblem : public FVProblem<TypeTag>
+{
+```
+
+<details><summary> Click to show alias definitions and local variables</summary>
+
+```cpp
+    using ParentType = FVProblem<TypeTag>;
+    using GridGeometry = GetPropType<TypeTag, Properties::GridGeometry>;
+    using FVElementGeometry = typename GridGeometry::LocalView;
+    using SubControlVolume = typename GridGeometry::SubControlVolume;
+    using GridView = typename GetPropType<TypeTag, Properties::GridGeometry>::GridView;
+    using Element = typename GridView::template Codim<0>::Entity;
+    using GlobalPosition = typename Element::Geometry::GlobalCoordinate;
+
+    using Scalar = GetPropType<TypeTag, Properties::Scalar>;
+    using PrimaryVariables = GetPropType<TypeTag, Properties::PrimaryVariables>;
+    using NumEqVector = Dumux::NumEqVector<PrimaryVariables>;
+    using BoundaryTypes = Dumux::BoundaryTypes<GetPropType<TypeTag, Properties::ModelTraits>::numEq()>;
+    using Indices = typename GetPropType<TypeTag, Properties::ModelTraits>::Indices;
+```
+
+</details>
+
+```cpp
+public:
+    CahnHilliardTestProblem(std::shared_ptr<const GridGeometry> gridGeometry)
+    : ParentType(gridGeometry)
+    {
+        mobility_ = getParam<Scalar>("Problem.Mobility");
+        surfaceTension_ = getParam<Scalar>("Problem.SurfaceTension");
+        energyScale_ = getParam<Scalar>("Problem.EnergyScale");
+    }
+```
+
+
+### Problem source term
+
+Here we implement the derivative of the free energy, setting a source for the equation for
+the chemical potential. The `computeSource` function in the local residual adds the terms
+defined here.
+
+
+```cpp
+    template<class ElementVolumeVariables>
+    NumEqVector source(const Element &element,
+                       const FVElementGeometry& fvGeometry,
+                       const ElementVolumeVariables& elemVolVars,
+                       const SubControlVolume &scv) const
+    {
+        NumEqVector values(0.0);
+        const auto& c = elemVolVars[scv].concentration();
+        values[Indices::chemicalPotentialEqIdx] = -energyScale_*2.0*c*(2.0*c*c - 3*c + 1);
+        return values;
+    }
+```
+
+
+### Boundary conditions
+
+For the boundary we choose boundary flux (or Neumann) conditions for all equations and on
+every part of the boundary, specifying zero flux everywhere for both equations.
+
+
+```cpp
+    BoundaryTypes boundaryTypesAtPos(const GlobalPosition& globalPos) const
+    {
+        BoundaryTypes values;
+        values.setAllNeumann();
+        return values;
+    }
+
+    NumEqVector neumannAtPos(const GlobalPosition& globalPos) const
+    { return { 0.0, 0.0 }; }
+```
+
+
+<details><summary> Click to show coefficients and private variables</summary>
+The problem class offers access to the mobility and surface tension coefficients as read from
+the parameter file by default `params.input`.
+
+
+```cpp
+    Scalar mobility() const
+    { return mobility_; }
+
+    Scalar surfaceTension() const
+    { return surfaceTension_; }
+
+private:
+    Scalar mobility_;
+    Scalar surfaceTension_;
+    Scalar energyScale_;
+};
+
+} // end namespace Dumux
+```
+
+</details>
+
+</details>
+
+## Test case properties/traits
+Within the `Dumux::Properties` namespace we specialize properties and traits to the considered
+test case by using the test's TypeTag.
+
+
+<details open>
+<summary><b>Click to hide/show the file documentation</b> (or inspect the [source code](../main.cc))</summary>
+
+
+### Include headers
+
+
+```cpp
+// Include the grid to be used
+#include <dune/grid/yaspgrid.hh>
+// The header for the box discretization scheme
+#include <dumux/discretization/box.hh>
+// The model header including the model traits and properties
+#include "model.hh"
+```
+
+
+### TypeTag `CahnHilliardTest`
+
+We define a type tag for the test case, allowing us to further specify properties and traits. To
+use those set for the Cahn-Hilliard model we inherit from its type tag.
+
+
+```cpp
+namespace Dumux::Properties {
+
+// Inheriting properties of the Cahn-Hilliard model and the box finite volume discretization
+namespace TTag {
+struct CahnHilliardTest { using InheritsFrom = std::tuple<CahnHilliardModel, BoxModel>; };
+} // end namespace TTag
+```
+
+
+### Test properties
+
+We specify a grid to be used in the test, select our problem class and enable caching.
+
+
+```cpp
+// Set the grid type
+template<class TypeTag>
+struct Grid<TypeTag, TTag::CahnHilliardTest>
+{ using type = Dune::YaspGrid<2>; };
+
+// Select the problem class defined above
+template<class TypeTag>
+struct Problem<TypeTag, TTag::CahnHilliardTest>
+{ using type = CahnHilliardTestProblem<TypeTag>; };
+
+// Enable caching
+template<class TypeTag>
+struct EnableGridVolumeVariablesCache<TypeTag, TTag::CahnHilliardTest>
+{ static constexpr bool value = true; };
+
+template<class TypeTag>
+struct EnableGridFluxVariablesCache<TypeTag, TTag::CahnHilliardTest>
+{ static constexpr bool value = true; };
+
+template<class TypeTag>
+struct EnableGridGeometryCache<TypeTag, TTag::CahnHilliardTest>
+{ static constexpr bool value = true; };
+
+} // end namespace Dumux::Properties
+```
+
+
+</details>
+
+## The main program flow
+The main program flow in the `main` function is often the only content of `main.cc`. It sets up
+the simulation framework, initializes runtime parameters, creates the grid and storage vectors
+for the variables, primary and secondary. It specifies and constructs and assembler, which
+assembles the discretized residual and system matrix (Jacobian of the model residual), as well as
+linear and nonlinear solvers that solve the resulting linear system and handle the convergence of
+nonlinear iterations. The time loop controls the time stepping, with adaptive time step size in
+coordination with the nonlinear solver.
+
+
+<details open>
+<summary><b>Click to hide/show the file documentation</b> (or inspect the [source code](../main.cc))</summary>
+
+
+### Include headers
+
+
+```cpp
+// standard header to generate random initial data
+#include <random>
+// common DuMux header for parallelization
+#include <dumux/common/initialize.hh>
+// headers to use the property system and runtime parameters
+#include <dumux/common/properties.hh>
+#include <dumux/common/parameters.hh>
+// module for VTK output, to write out fields of interest
+#include <dumux/io/vtkoutputmodule.hh>
+// gridmanager for the grid used in the test
+#include <dumux/io/grid/gridmanager_yasp.hh>
+// headers for linear and non-linear solvers as well as the assembler
+#include <dumux/linear/linearsolvertraits.hh>
+#include <dumux/linear/linearalgebratraits.hh>
+#include <dumux/linear/istlsolvers.hh>
+#include <dumux/nonlinear/newtonsolver.hh>
+#include <dumux/assembly/fvassembler.hh>
+```
+
+
+### Creating the initial solution
+
+We define a helper struct and function to handle communication for parallel runs.
+For our initial conditions we create a random field of values around a mean of 0.42.
+The random values are created with an offset based on the processor rank for communication
+purposes, which is removed afterwards. For more information see the description of the
+diffusion example
+[examples/diffusion/doc/main.md](https://git.iws.uni-stuttgart.de/dumux-repositories/dumux/-/blob/master/examples/diffusion/doc/main.md)
+
+
+```cpp
+struct MinScatter
+{
+    template<class A, class B>
+    static void apply(A& a, const B& b)
+    { a[0] = std::min(a[0], b[0]); }
+};
+
+template<class SolutionVector, class GridGeometry>
+SolutionVector createInitialSolution(const GridGeometry& gg)
+{
+    SolutionVector sol(gg.numDofs());
+
+    // Generate random number and add processor offset
+    // For sequential runs `rank` always returns `0`.
+    std::mt19937 gen(0); // seed is 0 for deterministic results
+    std::uniform_real_distribution<> dis(0.0, 1.0);
+    for (int n = 0; n < sol.size(); ++n)
+    {
+        sol[n][0] = 0.42 + 0.02*(0.5-dis(gen)) + gg.gridView().comm().rank();
+        sol[n][1] = 0.0;
+    }
+
+    // Take the value of the processor with the minimum rank and subtract the rank offset
+    if (gg.gridView().comm().size() > 1)
+    {
+        Dumux::VectorCommDataHandle<
+            typename GridGeometry::VertexMapper,
+            SolutionVector,
+            GridGeometry::GridView::dimension,
+            MinScatter
+        > minHandle(gg.vertexMapper(), sol);
+        gg.gridView().communicate(minHandle, Dune::All_All_Interface, Dune::ForwardCommunication);
+
+        // Remove processor offset
+        for (int n = 0; n < sol.size(); ++n)
+            sol[n][0] -= std::floor(sol[n][0]);
+    }
+    return sol;
+}
+```
+
+
+### The main function
+
+The main function takes the command line arguments, optionally specifying an input file of
+parameters and/or individual key-value pairs of runtime parameters.
+
+
+```cpp
+int main(int argc, char** argv)
+{
+    using namespace Dumux;
+
+    // define the type tag for this problem
+    using TypeTag = Properties::TTag::CahnHilliardTest;
+```
+
+
+We initialize parallelization backends as well as runtime parameters
+
+
+```cpp
+    // maybe initialize MPI and/or multithreading backend
+    Dumux::initialize(argc, argv);
+
+    // initialize parameter tree
+    Parameters::init(argc, argv);
+```
+
+
+### Grid setup
+
+Set up the grid as well as a grid geometry to access the (sub-)control-volumes and their
+faces
+
+
+```cpp
+    // initialize the grid
+    GridManager<GetPropType<TypeTag, Properties::Grid>> gridManager;
+    gridManager.init();
+
+    // we compute on the leaf grid view
+    const auto& leafGridView = gridManager.grid().leafGridView();
+
+    // create the finite volume grid geometry
+    using GridGeometry = GetPropType<TypeTag, Properties::GridGeometry>;
+    auto gridGeometry = std::make_shared<GridGeometry>(leafGridView);
+```
+
+
+### Problem setup
+
+We instantiate also the problem class according to the test properties
+
+
+```cpp
+    // the problem (initial and boundary conditions)
+    using Problem = GetPropType<TypeTag, Properties::Problem>;
+    auto problem = std::make_shared<Problem>(gridGeometry);
+```
+
+
+### Applying initial conditions
+
+After writing the initial data to the storage for previous and current time-step, we
+initialize the grid variables, also computing secondary variables.
+
+
+```cpp
+    // the solution vector
+    using SolutionVector = GetPropType<TypeTag, Properties::SolutionVector>;
+    auto sol = createInitialSolution<SolutionVector>(*gridGeometry);
+    // copy the vector to store state of previous time step
+    auto oldSol = sol;
+
+    // the grid variables
+    using GridVariables = GetPropType<TypeTag, Properties::GridVariables>;
+    auto gridVariables = std::make_shared<GridVariables>(problem, gridGeometry);
+    gridVariables->init(sol);
+```
+
+
+### Initialize VTK output
+
+
+```cpp
+    // initialize the vtk output module
+    VtkOutputModule<GridVariables, SolutionVector> vtkWriter(*gridVariables, sol, problem->name());
+    vtkWriter.addVolumeVariable([](const auto& vv){ return vv.concentration(); }, "c");
+    vtkWriter.addVolumeVariable([](const auto& vv){ return vv.chemicalPotential(); }, "mu");
+    vtkWriter.write(0.0);
+```
+
+
+### Set up time loop
+
+
+```cpp
+    // get some time loop parameters
+    using Scalar = GetPropType<TypeTag, Properties::Scalar>;
+    const auto tEnd = getParam<Scalar>("TimeLoop.TEnd");
+    const auto dt = getParam<Scalar>("TimeLoop.InitialTimeStepSize");
+    const auto maxDt = getParam<Scalar>("TimeLoop.MaxTimeStepSize");
+
+    // instantiate time loop
+    auto timeLoop = std::make_shared<CheckPointTimeLoop<Scalar>>(0.0, dt, tEnd);
+    timeLoop->setMaxTimeStepSize(maxDt);
+```
+
+
+### Assembler, linear and nonlinear solver
+
+
+```cpp
+    // the assembler with time loop for a transient problem
+    using Assembler = FVAssembler<TypeTag, DiffMethod::numeric>;
+    auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, oldSol);
+
+    // the linear solver
+    using LinearSolver = SSORBiCGSTABIstlSolver<
+        LinearSolverTraits<GridGeometry>,
+        LinearAlgebraTraitsFromAssembler<Assembler>
+    >;
+    auto linearSolver = std::make_shared<LinearSolver>(gridGeometry->gridView(), gridGeometry->dofMapper());
+
+    // the solver
+    using Solver = Dumux::NewtonSolver<Assembler, LinearSolver>;
+    Solver solver(assembler, linearSolver);
+```
+
+
+### Time loop
+
+
+```cpp
+    // time loop
+    timeLoop->start(); do
+    {
+        // assemble & solve
+        solver.solve(sol, *timeLoop);
+
+        // make the new solution the old solution
+        oldSol = sol;
+        gridVariables->advanceTimeStep();
+
+        // advance to the time loop to the next step
+        timeLoop->advanceTimeStep();
+
+        // write VTK output
+        vtkWriter.write(timeLoop->time());
+
+        // report statistics of this time step
+        timeLoop->reportTimeStep();
+
+        // set new dt as suggested by the Newton solver
+        timeLoop->setTimeStepSize(solver.suggestTimeStepSize(timeLoop->timeStepSize()));
+
+    } while (!timeLoop->finished());
+```
+
+
+### Finalize
+
+
+```cpp
+    timeLoop->finalize(leafGridView.comm());
+
+    return 0;
+}
+```
+
+
+</details>
+
+
+| [:arrow_left: Back to the main documentation](../README.md) | [:arrow_left: Go back to part 1](modelheader.md) |
+|---|---:|
+
diff --git a/examples/cahn_hilliard/doc/mainfile_intro.md b/examples/cahn_hilliard/doc/mainfile_intro.md
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+<!-- Important: This file has been automatically generated by generate_example_docs.py. Do not edit this file directly! -->
+
+
+| [:arrow_left: Back to the main documentation](../README.md) | [:arrow_right: Continue with part 2](mainfile.md) |
+|---|---:|
+
+
+
+# Volume variables, local residual and model traits (`model.hh`)
+In this example the file `model.hh` contains the classes `CahnHilliardModelVolumeVariables` and
+`CahnHilliardModelLocalResidual` as well as general model traits and properties.
+
+## VolumeVariables
+
+The volume variables store the local element volume variables, both primary and secondary.
+
+
+<details open>
+<summary><b>Click to hide/show the file documentation</b> (or inspect the [source code](../model.hh))</summary>
+
+
+### The `CahnHilliardModelVolumeVariables` class
+
+
+```cpp
+namespace Dumux {
+
+template <class Traits>
+class CahnHilliardModelVolumeVariables
+{
+    using Scalar = typename Traits::PrimaryVariables::value_type;
+    static_assert(Traits::PrimaryVariables::dimension == Traits::ModelTraits::numEq());
+public:
+    //! export the type used for the primary variables
+    using PrimaryVariables = typename Traits::PrimaryVariables;
+    //! export the indices type
+    using Indices = typename Traits::ModelTraits::Indices;
+```
+
+
+### Update variables
+
+The `update` function stores the local primary variables of the current solution and
+potentially recomputes secondary variables.
+
+
+```cpp
+    /*!
+     * \brief Update all quantities for a given control volume
+     */
+    template<class ElementSolution, class Problem, class Element, class SubControlVolume>
+    void update(const ElementSolution& elemSol,
+                const Problem& problem,
+                const Element& element,
+                const SubControlVolume& scv)
+    {
+        priVars_ = elemSol[scv.indexInElement()];
+    }
+```
+
+
+### Access functions
+
+Named and generic functions to access different primary variables
+
+
+```cpp
+    Scalar concentration() const
+    { return priVars_[Indices::concentrationIdx]; }
+
+    Scalar chemicalPotential() const
+    { return priVars_[Indices::chemicalPotentialIdx]; }
+
+    Scalar priVar(const int pvIdx) const
+    { return priVars_[pvIdx]; }
+
+    const PrimaryVariables& priVars() const
+    { return priVars_; }
+```
+
+
+### Extrusion factor
+
+The volumevariables are also expected to return the extrusion factor
+
+
+```cpp
+    // for compatibility with more general models
+    Scalar extrusionFactor() const
+    { return 1.0; }
+```
+
+
+### Storage of local variables
+
+
+```cpp
+private:
+    PrimaryVariables priVars_;
+};
+
+} // end namespace Dumux
+```
+
+
+</details>
+
+## LocalResidual
+
+The local residual defines the discretized and integrated partial differential equation through
+terms for storage, fluxes and sources, with the residual given as
+d/dt storage + div(fluxes) - sources = 0
+The individual terms can be further adjusted or replaced in the more specific problem.
+
+
+<details open>
+<summary><b>Click to hide/show the file documentation</b> (or inspect the [source code](../model.hh))</summary>
+
+
+### Include headers
+
+
+```cpp
+#include <dumux/common/math.hh>
+// use the property system
+#include <dumux/common/properties.hh>
+// common DuMux vector for discretized equations
+#include <dumux/common/numeqvector.hh>
+```
+
+
+### The local residual class `CahnHilliardModelLocalResidual` inherits from a base class set in
+the model properties, depending on the discretization scheme.
+
+
+```cpp
+namespace Dumux {
+
+template<class TypeTag>
+class CahnHilliardModelLocalResidual
+: public GetPropType<TypeTag, Properties::BaseLocalResidual>
+{
+    // the base local residual is selected depending on the chosen discretization scheme
+    using ParentType = GetPropType<TypeTag, Properties::BaseLocalResidual>;
+```
+
+
+<details><summary> Click to show alias definitions</summary>
+
+```cpp
+    using Scalar = GetPropType<TypeTag, Properties::Scalar>;
+    using Problem = GetPropType<TypeTag, Properties::Problem>;
+
+    using NumEqVector = Dumux::NumEqVector<GetPropType<TypeTag, Properties::PrimaryVariables>>;
+    using VolumeVariables = typename GetPropType<TypeTag, Properties::GridVolumeVariables>::VolumeVariables;
+
+    using ElementVolumeVariables = typename GetPropType<TypeTag, Properties::GridVolumeVariables>::LocalView;
+    using ElementFluxVariablesCache = typename GetPropType<TypeTag, Properties::GridFluxVariablesCache>::LocalView;
+    using FVElementGeometry = typename GetPropType<TypeTag, Properties::GridGeometry>::LocalView;
+
+    using SubControlVolume = typename FVElementGeometry::SubControlVolume;
+    using SubControlVolumeFace = typename FVElementGeometry::SubControlVolumeFace;
+    using GridView = typename GetPropType<TypeTag, Properties::GridGeometry>::GridView;
+    using Element = typename GridView::template Codim<0>::Entity;
+
+    using ModelTraits = GetPropType<TypeTag, Properties::ModelTraits>;
+    using Indices = typename ModelTraits::Indices;
+    static constexpr int dimWorld = GridView::dimensionworld;
+public:
+    using ParentType::ParentType;
+```
+
+</details>
+
+### The storage term
+The function `computeStorage` receives the volumevariables at the previous or current time
+step and computes the value of the storage terms.
+In this case the mass balance equation is a conservation equation of the concentration and
+the equation for the chemical potential does not have a storage term.
+
+
+```cpp
+    /*!
+     * \brief Evaluate the rate of change of all conserved quantities
+     */
+    NumEqVector computeStorage(const Problem& problem,
+                               const SubControlVolume& scv,
+                               const VolumeVariables& volVars) const
+    {
+        NumEqVector storage;
+        storage[Indices::massBalanceEqIdx] = volVars.concentration();
+        storage[Indices::chemicalPotentialEqIdx] = 0.0;
+        return storage;
+    }
+```
+
+
+### The flux terms
+`computeFlux` computes the fluxes over a subcontrolvolumeface, including the integration over
+the area of the face.
+
+
+```cpp
+    /*!
+     * \brief Evaluate the fluxes over a face of a sub control volume
+     * Here we evaluate the flow rate, F1 = -M∇mu·n A, F2 = -gamma∇c·n A
+     *
+     * TODO: why is this called flux, if we expect it to be integrated already?
+     * computeFluxIntegral?
+     */
+    NumEqVector computeFlux(const Problem& problem,
+                            const Element& element,
+                            const FVElementGeometry& fvGeometry,
+                            const ElementVolumeVariables& elemVolVars,
+                            const SubControlVolumeFace& scvf,
+                            const ElementFluxVariablesCache& elemFluxVarsCache) const
+    {
+        const auto& fluxVarCache = elemFluxVarsCache[scvf];
+        Dune::FieldVector<Scalar, dimWorld> gradConcentration(0.0);
+        Dune::FieldVector<Scalar, dimWorld> gradChemicalPotential(0.0);
+        for (const auto& scv : scvs(fvGeometry))
+        {
+            const auto& volVars = elemVolVars[scv];
+            gradConcentration.axpy(volVars.concentration(), fluxVarCache.gradN(scv.indexInElement()));
+            gradChemicalPotential.axpy(volVars.chemicalPotential(), fluxVarCache.gradN(scv.indexInElement()));
+        }
+
+        const auto M = problem.mobility();
+        const auto gamma = problem.surfaceTension();
+
+        NumEqVector flux;
+        flux[Indices::massBalanceEqIdx] = -1.0*vtmv(scvf.unitOuterNormal(), M, gradChemicalPotential)*scvf.area();
+        flux[Indices::chemicalPotentialEqIdx] = -1.0*vtmv(scvf.unitOuterNormal(), gamma, gradConcentration)*scvf.area();
+        return flux;
+    }
+```
+
+
+### Source terms
+
+`computeSource` defines the sources terms at a sub control volume.
+We implement a model-specific source term for the chemical potential equation before
+deferring further implementation to the problem where we add the derivative of the free
+energy. The default implementation of this function also defers the calculation to the
+problem.
+
+
+```cpp
+    /*!
+     * \brief Calculate the source term of the equation
+     */
+    NumEqVector computeSource(const Problem& problem,
+                              const Element& element,
+                              const FVElementGeometry& fvGeometry,
+                              const ElementVolumeVariables& elemVolVars,
+                              const SubControlVolume &scv) const
+    {
+        NumEqVector source(0.0);
+
+        source[Indices::chemicalPotentialEqIdx] = elemVolVars[scv].chemicalPotential();
+
+        // add contributions from problem (e.g. double well potential)
+        source += problem.source(element, fvGeometry, elemVolVars, scv);
+
+        return source;
+    }
+};
+
+} // end namespace Dumux
+```
+
+
+</details>
+
+## Model properties/traits
+
+We set some general model traits and properties, using a TypeTag `CahnHilliardModel`.
+For this type tag we define a `ModelTraits` struct and set a number of properties by specifying
+further structs within the `Dumux::Properties` namespace.
+
+
+<details open>
+<summary><b>Click to hide/show the file documentation</b> (or inspect the [source code](../model.hh))</summary>
+
+
+### Include the header for the property system and common properties and switch to the
+`Properties` namespace.
+
+
+```cpp
+#include <dumux/common/properties.hh>
+
+namespace Dumux::Properties {
+```
+
+
+### Define the type tag
+
+
+```cpp
+namespace TTag {
+struct CahnHilliardModel {};
+} // end namespace TTag
+```
+
+
+### Basic model properties
+
+Define some general properties to be used for this modedl such as a datatype for scalars and the
+default vector for the primary variables. Here we can also use the model traits we specify below.
+
+
+```cpp
+//! Set the default type of scalar values to double
+template<class TypeTag>
+struct Scalar<TypeTag, TTag:: CahnHilliardModel >
+{ using type = double; };
+
+//! Set the default primary variable vector to a vector of size of number of equations
+template<class TypeTag>
+struct PrimaryVariables<TypeTag, TTag:: CahnHilliardModel >
+{
+    using type = Dune::FieldVector<
+        GetPropType<TypeTag, Properties::Scalar>,
+        GetPropType<TypeTag, Properties::ModelTraits>::numEq()
+    >;
+};
+```
+
+
+### Model traits
+
+We specify general traits of the implemented model, defining indices (often in `indices.hh`)
+and the number of equations in the model.
+
+
+```cpp
+//! Set the model traits property
+template<class TypeTag>
+struct ModelTraits<TypeTag, TTag::CahnHilliardModel>
+{
+    struct Traits
+    {
+        struct Indices
+        {
+            static constexpr int concentrationIdx = 0;
+            static constexpr int chemicalPotentialIdx = 1;
+
+            static constexpr int massBalanceEqIdx = 0;
+            static constexpr int chemicalPotentialEqIdx = 1;
+        };
+
+        static constexpr int numEq() { return 2; }
+    };
+
+    using type = Traits;
+};
+```
+
+
+### Further model properties
+
+Define further properties of the model, selecting the local residual and volumevariables defined
+above.
+
+
+```cpp
+template<class TypeTag>
+struct LocalResidual<TypeTag, TTag::CahnHilliardModel>
+{ using type = CahnHilliardModelLocalResidual<TypeTag>; };
+
+//! Set the volume variables property
+template<class TypeTag>
+struct VolumeVariables<TypeTag, TTag::CahnHilliardModel>
+{
+    struct Traits
+    {
+        using PrimaryVariables = GetPropType<TypeTag, Properties::PrimaryVariables>;
+        using ModelTraits = GetPropType<TypeTag, Properties::ModelTraits>;
+    };
+
+    using type = CahnHilliardModelVolumeVariables<Traits>;
+};
+
+} // end namespace Dumux::Properties
+```
+
+
+</details>
+
+
+| [:arrow_left: Back to the main documentation](../README.md) | [:arrow_right: Continue with part 2](mainfile.md) |
+|---|---:|
+
diff --git a/examples/cahn_hilliard/doc/modelheader_intro.md b/examples/cahn_hilliard/doc/modelheader_intro.md
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HcmV?d00001

diff --git a/examples/cahn_hilliard/main.cc b/examples/cahn_hilliard/main.cc
index fa1bc08904..4e18dcff99 100644
--- a/examples/cahn_hilliard/main.cc
+++ b/examples/cahn_hilliard/main.cc
@@ -17,26 +17,43 @@
  *   along with this program.  If not, see <http://www.gnu.org/licenses/>.   *
  *****************************************************************************/
 
+// # Problem, test properties/traits and main program flow (`main.cc`)
+// In this example the file `main.cc` contains the problem class `CahnHilliardTestProblem`,
+// properties and traits specific to the test case as well as the main program flow in the form of
+// `main` function.
+//
 #include <config.h>
-
-//////////////////////////////////////////////////////////////////////////////
-//////////////////////////////////////////////////////////////////////////////
-// Problem //////// (often in a separate file problem.hh) ////////////////////
-//////////////////////////////////////////////////////////////////////////////
-//////////////////////////////////////////////////////////////////////////////
-
+// ## Problem
+// The __problem class__ defines boundary conditions and extends the storage term defined in the
+// model's localresidual by the derivative of the free energy.
+//
+// [[content]]
+// ### Include headers
+//
+// [[codeblock]]
+// use the property system and runtime parameters
 #include <dumux/common/properties.hh>
 #include <dumux/common/parameters.hh>
+// common DuMux vector for discretized equations
 #include <dumux/common/numeqvector.hh>
-
+// types of boundary conditions to use
 #include <dumux/common/boundarytypes.hh>
+// generic problem for finite volume simulations
 #include <dumux/common/fvproblem.hh>
-
+// [[/codeblock]]
+//
+// ### The problem class `CahnHilliardTestProblem`
+// In this abstract problem we inherit from the generic `FVProblem`.
+//
+// [[codeblock]]
 namespace Dumux {
 
 template<class TypeTag>
 class CahnHilliardTestProblem : public FVProblem<TypeTag>
 {
+// [[/codeblock]]
+// [[details]] alias definitions and local variables
+// [[codeblock]]
     using ParentType = FVProblem<TypeTag>;
     using GridGeometry = GetPropType<TypeTag, Properties::GridGeometry>;
     using FVElementGeometry = typename GridGeometry::LocalView;
@@ -50,6 +67,9 @@ class CahnHilliardTestProblem : public FVProblem<TypeTag>
     using NumEqVector = Dumux::NumEqVector<PrimaryVariables>;
     using BoundaryTypes = Dumux::BoundaryTypes<GetPropType<TypeTag, Properties::ModelTraits>::numEq()>;
     using Indices = typename GetPropType<TypeTag, Properties::ModelTraits>::Indices;
+// [[/codeblock]]
+// [[/details]]
+// [[codeblock]]
 public:
     CahnHilliardTestProblem(std::shared_ptr<const GridGeometry> gridGeometry)
     : ParentType(gridGeometry)
@@ -58,8 +78,15 @@ public:
         surfaceTension_ = getParam<Scalar>("Problem.SurfaceTension");
         energyScale_ = getParam<Scalar>("Problem.EnergyScale");
     }
-
-    // here we implement the derivative of the free energy
+    // [[/codeblock]]
+    //
+    // ### Problem source term
+    //
+    // Here we implement the derivative of the free energy, setting a source for the equation for
+    // the chemical potential. The `computeSource` function in the local residual adds the terms
+    // defined here.
+    //
+    // [[codeblock]]
     template<class ElementVolumeVariables>
     NumEqVector source(const Element &element,
                        const FVElementGeometry& fvGeometry,
@@ -71,7 +98,14 @@ public:
         values[Indices::chemicalPotentialEqIdx] = -energyScale_*2.0*c*(2.0*c*c - 3*c + 1);
         return values;
     }
-
+    // [[/codeblock]]
+    //
+    // ### Boundary conditions
+    //
+    // For the boundary we choose boundary flux (or Neumann) conditions for all equations and on
+    // every part of the boundary, specifying zero flux everywhere for both equations.
+    //
+    // [[codeblock]]
     BoundaryTypes boundaryTypesAtPos(const GlobalPosition& globalPos) const
     {
         BoundaryTypes values;
@@ -81,7 +115,13 @@ public:
 
     NumEqVector neumannAtPos(const GlobalPosition& globalPos) const
     { return { 0.0, 0.0 }; }
-
+    // [[/codeblock]]
+    //
+    // [[details]] coefficients and private variables
+    // The problem class offers access to the mobility and surface tension coefficients as read from
+    // the parameter file by default `params.input`.
+    //
+    // [[codeblock]]
     Scalar mobility() const
     { return mobility_; }
 
@@ -95,29 +135,52 @@ private:
 };
 
 } // end namespace Dumux
-
-//////////////////////////////////////////////////////////////////////////////
-
-//////////////////////////////////////////////////////////////////////////////
-//////////////////////////////////////////////////////////////////////////////
-// Test case properties/traits /// (often in a separate file properties.hh) //
-//////////////////////////////////////////////////////////////////////////////
-//////////////////////////////////////////////////////////////////////////////
-
+// [[/codeblock]]
+// [[/details]]
+// [[/content]]
+//
+// ## Test case properties/traits
+// Within the `Dumux::Properties` namespace we specialize properties and traits to the considered
+// test case by using the test's TypeTag.
+//
+// [[content]]
+//
+// ### Include headers
+//
+// [[codeblock]]
+// Include the grid to be used
 #include <dune/grid/yaspgrid.hh>
+// The header for the box discretization scheme
 #include <dumux/discretization/box.hh>
+// The model header including the model traits and properties
 #include "model.hh"
-
+// [[/codeblock]]
+//
+// ### TypeTag `CahnHilliardTest`
+//
+// We define a type tag for the test case, allowing us to further specify properties and traits. To
+// use those set for the Cahn-Hilliard model we inherit from its type tag.
+//
+// [[codeblock]]
 namespace Dumux::Properties {
 
+// Inheriting properties of the Cahn-Hilliard model and the box finite volume discretization
 namespace TTag {
 struct CahnHilliardTest { using InheritsFrom = std::tuple<CahnHilliardModel, BoxModel>; };
 } // end namespace TTag
-
+// [[/codeblock]]
+//
+// ### Test properties
+//
+// We specify a grid to be used in the test, select our problem class and enable caching.
+//
+// [[codeblock]]
+// Set the grid type
 template<class TypeTag>
 struct Grid<TypeTag, TTag::CahnHilliardTest>
 { using type = Dune::YaspGrid<2>; };
 
+// Select the problem class defined above
 template<class TypeTag>
 struct Problem<TypeTag, TTag::CahnHilliardTest>
 { using type = CahnHilliardTestProblem<TypeTag>; };
@@ -136,24 +199,52 @@ struct EnableGridGeometryCache<TypeTag, TTag::CahnHilliardTest>
 { static constexpr bool value = true; };
 
 } // end namespace Dumux::Properties
-
-//////////////////////////////////////////////////////////////////////////////
-
+// [[/codeblock]]
+// [[/content]]
+//
+// ## The main program flow
+// The main program flow in the `main` function is often the only content of `main.cc`. It sets up
+// the simulation framework, initializes runtime parameters, creates the grid and storage vectors
+// for the variables, primary and secondary. It specifies and constructs and assembler, which
+// assembles the discretized residual and system matrix (Jacobian of the model residual), as well as
+// linear and nonlinear solvers that solve the resulting linear system and handle the convergence of
+// nonlinear iterations. The time loop controls the time stepping, with adaptive time step size in
+// coordination with the nonlinear solver.
+//
+// [[content]]
+//
+// ### Include headers
+//
+// [[codeblock]]
+// standard header to generate random initial data
 #include <random>
-
+// common DuMux header for parallelization
 #include <dumux/common/initialize.hh>
+// headers to use the property system and runtime parameters
 #include <dumux/common/properties.hh>
 #include <dumux/common/parameters.hh>
-
+// module for VTK output, to write out fields of interest
 #include <dumux/io/vtkoutputmodule.hh>
+// gridmanager for the grid used in the test
 #include <dumux/io/grid/gridmanager_yasp.hh>
-
+// headers for linear and non-linear solvers as well as the assembler
 #include <dumux/linear/linearsolvertraits.hh>
 #include <dumux/linear/linearalgebratraits.hh>
 #include <dumux/linear/istlsolvers.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
 #include <dumux/assembly/fvassembler.hh>
-
+// [[/codeblock]]
+//
+// ### Creating the initial solution
+//
+// We define a helper struct and function to handle communication for parallel runs.
+// For our initial conditions we create a random field of values around a mean of 0.42.
+// The random values are created with an offset based on the processor rank for communication
+// purposes, which is removed afterwards. For more information see the description of the
+// diffusion example
+// [examples/diffusion/doc/main.md](https://git.iws.uni-stuttgart.de/dumux-repositories/dumux/-/blob/master/examples/diffusion/doc/main.md)
+//
+// [[codeblock]]
 struct MinScatter
 {
     template<class A, class B>
@@ -161,19 +252,70 @@ struct MinScatter
     { a[0] = std::min(a[0], b[0]); }
 };
 
+template<class SolutionVector, class GridGeometry>
+SolutionVector createInitialSolution(const GridGeometry& gg)
+{
+    SolutionVector sol(gg.numDofs());
+
+    // Generate random number and add processor offset
+    // For sequential runs `rank` always returns `0`.
+    std::mt19937 gen(0); // seed is 0 for deterministic results
+    std::uniform_real_distribution<> dis(0.0, 1.0);
+    for (int n = 0; n < sol.size(); ++n)
+    {
+        sol[n][0] = 0.42 + 0.02*(0.5-dis(gen)) + gg.gridView().comm().rank();
+        sol[n][1] = 0.0;
+    }
+
+    // Take the value of the processor with the minimum rank and subtract the rank offset
+    if (gg.gridView().comm().size() > 1)
+    {
+        Dumux::VectorCommDataHandle<
+            typename GridGeometry::VertexMapper,
+            SolutionVector,
+            GridGeometry::GridView::dimension,
+            MinScatter
+        > minHandle(gg.vertexMapper(), sol);
+        gg.gridView().communicate(minHandle, Dune::All_All_Interface, Dune::ForwardCommunication);
+
+        // Remove processor offset
+        for (int n = 0; n < sol.size(); ++n)
+            sol[n][0] -= std::floor(sol[n][0]);
+    }
+    return sol;
+}
+// [[/codeblock]]
+//
+// ### The main function
+//
+// The main function takes the command line arguments, optionally specifying an input file of
+// parameters and/or individual key-value pairs of runtime parameters.
+//
+// [[codeblock]]
 int main(int argc, char** argv)
 {
     using namespace Dumux;
 
     // define the type tag for this problem
     using TypeTag = Properties::TTag::CahnHilliardTest;
-
+    // [[/codeblock]]
+    //
+    // We initialize parallelization backends as well as runtime parameters
+    //
+    // [[codeblock]]
     // maybe initialize MPI and/or multithreading backend
     Dumux::initialize(argc, argv);
 
     // initialize parameter tree
     Parameters::init(argc, argv);
-
+    // [[/codeblock]]
+    //
+    // ### Grid setup
+    //
+    // Set up the grid as well as a grid geometry to access the (sub-)control-volumes and their
+    // faces
+    //
+    // [[codeblock]]
     // initialize the grid
     GridManager<GetPropType<TypeTag, Properties::Grid>> gridManager;
     gridManager.init();
@@ -184,37 +326,27 @@ int main(int argc, char** argv)
     // create the finite volume grid geometry
     using GridGeometry = GetPropType<TypeTag, Properties::GridGeometry>;
     auto gridGeometry = std::make_shared<GridGeometry>(leafGridView);
-
+    // [[/codeblock]]
+    //
+    // ### Problem setup
+    //
+    // We instantiate also the problem class according to the test properties
+    //
+    // [[codeblock]]
     // the problem (initial and boundary conditions)
     using Problem = GetPropType<TypeTag, Properties::Problem>;
     auto problem = std::make_shared<Problem>(gridGeometry);
-
+    // [[/codeblock]]
+    //
+    // ### Applying initial conditions
+    //
+    // After writing the initial data to the storage for previous and current time-step, we
+    // initialize the grid variables, also computing secondary variables.
+    //
+    // [[codeblock]]
     // the solution vector
     using SolutionVector = GetPropType<TypeTag, Properties::SolutionVector>;
-    SolutionVector sol(gridGeometry->numDofs());
-
-    // create a random initial field
-    std::mt19937 gen(0); // seed is 0 for deterministic results
-    std::uniform_real_distribution<> dis(0.0, 1.0);
-    for (int n = 0; n < sol.size(); ++n)
-    {
-        sol[n][0] = 0.42 + 0.02*(0.5-dis(gen)) + leafGridView.comm().rank();
-        sol[n][1] = 0.0;
-    }
-
-    // take the value of the processor with the minimum rank
-    VectorCommDataHandle<
-        typename GridGeometry::VertexMapper,
-        SolutionVector,
-        GridGeometry::GridView::dimension,
-        MinScatter
-    > minHandle(gridGeometry->vertexMapper(), sol);
-    leafGridView.communicate(minHandle, Dune::All_All_Interface, Dune::ForwardCommunication);
-
-    // remove processor offset
-    for (int n = 0; n < sol.size(); ++n)
-        sol[n][0] -= std::floor(sol[n][0]);
-
+    auto sol = createInitialSolution<SolutionVector>(*gridGeometry);
     // copy the vector to store state of previous time step
     auto oldSol = sol;
 
@@ -222,13 +354,21 @@ int main(int argc, char** argv)
     using GridVariables = GetPropType<TypeTag, Properties::GridVariables>;
     auto gridVariables = std::make_shared<GridVariables>(problem, gridGeometry);
     gridVariables->init(sol);
-
+    // [[/codeblock]]
+    //
+    // ### Initialize VTK output
+    //
+    // [[codeblock]]
     // initialize the vtk output module
     VtkOutputModule<GridVariables, SolutionVector> vtkWriter(*gridVariables, sol, problem->name());
     vtkWriter.addVolumeVariable([](const auto& vv){ return vv.concentration(); }, "c");
     vtkWriter.addVolumeVariable([](const auto& vv){ return vv.chemicalPotential(); }, "mu");
     vtkWriter.write(0.0);
-
+    // [[/codeblock]]
+    //
+    // ### Set up time loop
+    //
+    // [[codeblock]]
     // get some time loop parameters
     using Scalar = GetPropType<TypeTag, Properties::Scalar>;
     const auto tEnd = getParam<Scalar>("TimeLoop.TEnd");
@@ -238,7 +378,11 @@ int main(int argc, char** argv)
     // instantiate time loop
     auto timeLoop = std::make_shared<CheckPointTimeLoop<Scalar>>(0.0, dt, tEnd);
     timeLoop->setMaxTimeStepSize(maxDt);
-
+    // [[/codeblock]]
+    //
+    // ### Assembler, linear and nonlinear solver
+    //
+    // [[codeblock]]
     // the assembler with time loop for a transient problem
     using Assembler = FVAssembler<TypeTag, DiffMethod::numeric>;
     auto assembler = std::make_shared<Assembler>(problem, gridGeometry, gridVariables, timeLoop, oldSol);
@@ -253,7 +397,11 @@ int main(int argc, char** argv)
     // the solver
     using Solver = Dumux::NewtonSolver<Assembler, LinearSolver>;
     Solver solver(assembler, linearSolver);
-
+    // [[/codeblock]]
+    //
+    // ### Time loop
+    //
+    // [[codeblock]]
     // time loop
     timeLoop->start(); do
     {
@@ -277,8 +425,14 @@ int main(int argc, char** argv)
         timeLoop->setTimeStepSize(solver.suggestTimeStepSize(timeLoop->timeStepSize()));
 
     } while (!timeLoop->finished());
-
+    // [[/codeblock]]
+    //
+    // ### Finalize
+    //
+    // [[codeblock]]
     timeLoop->finalize(leafGridView.comm());
 
     return 0;
 }
+// [[/codeblock]]
+// [[/content]]
diff --git a/examples/cahn_hilliard/model.hh b/examples/cahn_hilliard/model.hh
index 51b5eb62be..9f69f9dada 100644
--- a/examples/cahn_hilliard/model.hh
+++ b/examples/cahn_hilliard/model.hh
@@ -20,12 +20,19 @@
 #ifndef DUMUX_EXAMPLES_CAHN_HILLIARD_MODEL_HH
 #define DUMUX_EXAMPLES_CAHN_HILLIARD_MODEL_HH
 
-//////////////////////////////////////////////////////////////////////////////
-//////////////////////////////////////////////////////////////////////////////
-// VolumeVariables //////// (often in a separate file volumevariables.hh) ////
-//////////////////////////////////////////////////////////////////////////////
-//////////////////////////////////////////////////////////////////////////////
-
+// # Volume variables, local residual and model traits (`model.hh`)
+// In this example the file `model.hh` contains the classes `CahnHilliardModelVolumeVariables` and
+// `CahnHilliardModelLocalResidual` as well as general model traits and properties.
+//
+// ## VolumeVariables
+//
+// The volume variables store the local element volume variables, both primary and secondary.
+//
+// [[content]]
+//
+// ### The `CahnHilliardModelVolumeVariables` class
+//
+// [[codeblock]]
 namespace Dumux {
 
 template <class Traits>
@@ -38,7 +45,14 @@ public:
     using PrimaryVariables = typename Traits::PrimaryVariables;
     //! export the indices type
     using Indices = typename Traits::ModelTraits::Indices;
-
+    // [[/codeblock]]
+    //
+    // ### Update variables
+    //
+    // The `update` function stores the local primary variables of the current solution and
+    // potentially recomputes secondary variables.
+    //
+    // [[codeblock]]
     /*!
      * \brief Update all quantities for a given control volume
      */
@@ -50,7 +64,13 @@ public:
     {
         priVars_ = elemSol[scv.indexInElement()];
     }
-
+    // [[/codeblock]]
+    //
+    // ### Access functions
+    //
+    // Named and generic functions to access different primary variables
+    //
+    // [[codeblock]]
     Scalar concentration() const
     { return priVars_[Indices::concentrationIdx]; }
 
@@ -62,29 +82,52 @@ public:
 
     const PrimaryVariables& priVars() const
     { return priVars_; }
-
+    // [[/codeblock]]
+    //
+    // ### Extrusion factor
+    //
+    // The volumevariables are also expected to return the extrusion factor
+    //
+    // [[codeblock]]
     // for compatibility with more general models
     Scalar extrusionFactor() const
     { return 1.0; }
-
+    // [[/codeblock]]
+    //
+    // ### Storage of local variables
+    //
+    // [[codeblock]]
 private:
     PrimaryVariables priVars_;
 };
 
 } // end namespace Dumux
-
-//////////////////////////////////////////////////////////////////////////////
-
-//////////////////////////////////////////////////////////////////////////////
-//////////////////////////////////////////////////////////////////////////////
-// LocalResidual //////////// (often in a separate file localresidual.hh) ////
-//////////////////////////////////////////////////////////////////////////////
-//////////////////////////////////////////////////////////////////////////////
-
+// [[/codeblock]]
+// [[/content]]
+//
+// ## LocalResidual
+//
+// The local residual defines the discretized and integrated partial differential equation through
+// terms for storage, fluxes and sources, with the residual given as
+// d/dt storage + div(fluxes) - sources = 0
+// The individual terms can be further adjusted or replaced in the more specific problem.
+//
+// [[content]]
+//
+// ### Include headers
+//
+// [[codeblock]]
 #include <dumux/common/math.hh>
+// use the property system
 #include <dumux/common/properties.hh>
+// common DuMux vector for discretized equations
 #include <dumux/common/numeqvector.hh>
-
+// [[/codeblock]]
+//
+// ### The local residual class `CahnHilliardModelLocalResidual` inherits from a base class set in
+// the model properties, depending on the discretization scheme.
+//
+// [[codeblock]]
 namespace Dumux {
 
 template<class TypeTag>
@@ -93,6 +136,10 @@ class CahnHilliardModelLocalResidual
 {
     // the base local residual is selected depending on the chosen discretization scheme
     using ParentType = GetPropType<TypeTag, Properties::BaseLocalResidual>;
+    // [[/codeblock]]
+    //
+    // [[details]] alias definitions
+    // [[codeblock]]
     using Scalar = GetPropType<TypeTag, Properties::Scalar>;
     using Problem = GetPropType<TypeTag, Properties::Problem>;
 
@@ -113,7 +160,16 @@ class CahnHilliardModelLocalResidual
     static constexpr int dimWorld = GridView::dimensionworld;
 public:
     using ParentType::ParentType;
-
+    // [[/codeblock]]
+    // [[/details]]
+    //
+    // ### The storage term
+    // The function `computeStorage` receives the volumevariables at the previous or current time
+    // step and computes the value of the storage terms.
+    // In this case the mass balance equation is a conservation equation of the concentration and
+    // the equation for the chemical potential does not have a storage term.
+    //
+    // [[codeblock]]
     /*!
      * \brief Evaluate the rate of change of all conserved quantities
      */
@@ -126,7 +182,13 @@ public:
         storage[Indices::chemicalPotentialEqIdx] = 0.0;
         return storage;
     }
-
+    // [[/codeblock]]
+    //
+    // ### The flux terms
+    // `computeFlux` computes the fluxes over a subcontrolvolumeface, including the integration over
+    // the area of the face.
+    //
+    // [[codeblock]]
     /*!
      * \brief Evaluate the fluxes over a face of a sub control volume
      * Here we evaluate the flow rate, F1 = -M∇mu·n A, F2 = -gamma∇c·n A
@@ -159,7 +221,17 @@ public:
         flux[Indices::chemicalPotentialEqIdx] = -1.0*vtmv(scvf.unitOuterNormal(), gamma, gradConcentration)*scvf.area();
         return flux;
     }
-
+    // [[/codeblock]]
+    //
+    // ### Source terms
+    //
+    // `computeSource` defines the sources terms at a sub control volume.
+    // We implement a model-specific source term for the chemical potential equation before
+    // deferring further implementation to the problem where we add the derivative of the free
+    // energy. The default implementation of this function also defers the calculation to the
+    // problem.
+    //
+    // [[codeblock]]
     /*!
      * \brief Calculate the source term of the equation
      */
@@ -181,22 +253,40 @@ public:
 };
 
 } // end namespace Dumux
-//////////////////////////////////////////////////////////////////////////////
-
-//////////////////////////////////////////////////////////////////////////////
-//////////////////////////////////////////////////////////////////////////////
-// Model properties/traits ///////////////////////////////////////////////////
-//////////////////////////////////////////////////////////////////////////////
-//////////////////////////////////////////////////////////////////////////////
-
+// [[/codeblock]]
+// [[/content]]
+//
+// ## Model properties/traits
+//
+// We set some general model traits and properties, using a TypeTag `CahnHilliardModel`.
+// For this type tag we define a `ModelTraits` struct and set a number of properties by specifying
+// further structs within the `Dumux::Properties` namespace.
+//
+// [[content]]
+//
+// ### Include the header for the property system and common properties and switch to the
+// `Properties` namespace.
+//
+// [[codeblock]]
 #include <dumux/common/properties.hh>
 
 namespace Dumux::Properties {
-
+// [[/codeblock]]
+//
+// ### Define the type tag
+//
+// [[codeblock]]
 namespace TTag {
 struct CahnHilliardModel {};
 } // end namespace TTag
-
+// [[/codeblock]]
+//
+// ### Basic model properties
+//
+// Define some general properties to be used for this modedl such as a datatype for scalars and the
+// default vector for the primary variables. Here we can also use the model traits we specify below.
+//
+// [[codeblock]]
 //! Set the default type of scalar values to double
 template<class TypeTag>
 struct Scalar<TypeTag, TTag:: CahnHilliardModel >
@@ -211,7 +301,14 @@ struct PrimaryVariables<TypeTag, TTag:: CahnHilliardModel >
         GetPropType<TypeTag, Properties::ModelTraits>::numEq()
     >;
 };
-
+// [[/codeblock]]
+//
+// ### Model traits
+//
+// We specify general traits of the implemented model, defining indices (often in `indices.hh`)
+// and the number of equations in the model.
+//
+// [[codeblock]]
 //! Set the model traits property
 template<class TypeTag>
 struct ModelTraits<TypeTag, TTag::CahnHilliardModel>
@@ -232,7 +329,14 @@ struct ModelTraits<TypeTag, TTag::CahnHilliardModel>
 
     using type = Traits;
 };
-
+// [[/codeblock]]
+//
+// ### Further model properties
+//
+// Define further properties of the model, selecting the local residual and volumevariables defined
+// above.
+//
+// [[codeblock]]
 template<class TypeTag>
 struct LocalResidual<TypeTag, TTag::CahnHilliardModel>
 { using type = CahnHilliardModelLocalResidual<TypeTag>; };
@@ -251,5 +355,7 @@ struct VolumeVariables<TypeTag, TTag::CahnHilliardModel>
 };
 
 } // end namespace Dumux::Properties
+// [[/codeblock]]
+// [[/content]]
 
 #endif
-- 
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