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Tutorials

On this page you find an overview of Ferrite tutorials. The tutorials explain and show how Ferrite can be used to solve a wide range of problems. See also the Code gallery for more examples.

The tutorials all follow roughly the same structure:

  • Introduction introduces the problem to be solved and discusses the learning outcomes of the tutorial.
  • Commented program is the code for solving the problem with explanations and comments.
  • Plain program is the raw source code of the program.

When studying the tutorials it is a good idea to obtain a local copy of the code and run it on your own machine as you read along. Some of the tutorials also include suggestions for tweaks to the program that you can try out on your own.

Tutorial index

The tutorials are listed in roughly increasing order of complexity. However, since they focus on different aspects, and solve different problems, it is suggested to have a look at the brief descriptions below to get an idea about what you will learn from each tutorial.

If you are new to Ferrite then Tutorial 1 - Tutorial 6 is the best place to start. These tutorials introduces and teaches most of the basic finite element techniques (e.g. linear and non-linear problems, scalar- and vector-valued problems, Dirichlet and Neumann boundary conditions, mixed finite elements, time integration, direct and iterative linear solvers, etc). In particular the very first tutorial is essential in order to be able to follow any of the other tutorials. The remaining tutorials discuss more advanced topics.

Tutorial 1: Heat equation

This tutorial guides you through the process of solving the linear stationary heat equation (i.e. Poisson's equation) on a unit square with homogeneous Dirichlet boundary conditions. This tutorial introduces and teaches many important parts of Ferrite: problem setup, degree of freedom management, assembly procedure, boundary conditions, solving the linear system, visualization of the result). Understanding this tutorial is essential to follow more complex tutorials.

Keywords: scalar-valued solution, Dirichlet boundary conditions.

Tutorial 2: Linear elasticity

TBW.

Keywords: vector-valued solution, Dirichlet and Neumann boundary conditions.

Tutorial 3: Incompressible elasticity

This tutorial focuses on a mixed formulation of linear elasticity, with (vector) displacement and (scalar) pressure as the two unknowns, suitable for incompressibility. Thus, this tutorial guides you through the process of solving a problem with two unknowns from two coupled weak forms. The problem that is studied is Cook's membrane in the incompressible limit.

Keywords: mixed finite elements, Dirichlet and Neumann boundary conditions.

Tutorial 4: Hyperelasticity

In this tutorial you will learn how to solve a non-linear finite element problem. In particular, a hyperelastic material model, in a finite strain setting, is used to solve the rotation of a cube. Automatic differentiatio (AD) is used for the consitutive relations. Newton's method is used for the non-linear iteration, and a conjugate gradient (CG) solver is used for the linear solution of the increment.

Keywords: non-linear finite element, finite strain, automatic differentiation (AD), Newton's method, conjugate gradient (CG).

Tutorial 5: von Mises Plasticity

This tutorial revisits the cantilever beam problem from Tutorial 2: Linear elasticity, but instead of linear elasticity a plasticity model is used for the constitutive relation. You will learn how to solve a problem which require the solution of a local material problem, and the storage of material state, in each quadrature point. Newton's method is used both locally in the material routine, and globally on the finite element level.

Keywords: non-linear finite element, plasticity, material modeling, state variables, Newton’s method.

Tutorial 6: Transient heat equation

In this tutorial the transient heat equation is solved on the unit square. The problem to be solved is thus similar to the one solved in the first tutorial, Heat equation, but with time-varying boundary conditions. In particular you will learn how to solve a time dependent problem with an implicit Euler scheme for the time integration.

Keywords: time dependent finite elements, implicit Euler time integration.

Tutorial 7: Computational homogenization

This tutorial guides you through computational homogenization of an representative volume element (RVE) consisting of a soft matrix material with stiff inclusions. The computational mesh is read from an external mesh file generated with Gmsh. Dirichlet and periodic boundary conditions are used.

Keywords: Gmsh mesh reading, Dirichlet and periodic boundary conditions

Tutorial 8: Stokes flow

In this tutorial Stokes flow with (vector) velocity and (scalar) pressure is solved on on a quarter circle. Rotationally periodic boundary conditions is used for the inlet/outlet coupling. To obtain a unique solution, a mean value constraint is applied on the pressure using an affine constraint. The computational mesh is generated directly using the Gmsh API.

Keywords: periodic boundary conditions, mean value constraint, mesh generation with Gmsh.

Tutorial 9: Porous media (SubDofHandler)

This tutorial introduces how to solve a complex linear problem, where there are different fields on different subdomains, and different cell types in the grid. This requires using the SubDofHandler interface.

Keywords: Mixed grids, multiple fields, porous media, SubDofHandler

Tutorial 10: Incompressible Navier-Stokes equations

In this tutorial the incompressible Navier-Stokes equations are solved. The domain is discretized in space with Ferrite as usual, and then forumalated in a way to be compatible with the OrdinaryDiffEq.jl package, which is used for the time-integration.

Keywords: non-linear time dependent problem

Tutorial 11: Linear shell

In this tutorial a linear shell element formulation is set up as a two-dimensional domain embedded in three-dimensional space. This will teach, and perhaps inspire, you on how Ferrite can be used for non-standard things and how to add "hacks" that build on top of Ferrite.

Keywords: embedding, automatic differentiation

Tutorial 11: Discontinuous Galerkin heat equation

This tutorial guides you through the process of solving the linear stationary heat equation (i.e. Poisson's equation) on a unit square with inhomogeneous Dirichlet and Neumann boundary conditions using the interior penalty discontinuous Galerkin method. This tutorial follows the heat equation tutorial, introducing face and interface iterators, jump and average operators, and cross-element coupling in sparsity patterns. This example was developed as part of the Google Summer of Code funded project "Discontinuous Galerkin Infrastructure For the finite element toolbox Ferrite.jl".

Keywords: scalar-valued solution, Dirichlet boundary conditions, Discontinuous Galerkin, Interior penalty.