Fenics documentation

In this tour, we will solve a transient thermoelastic evolution problem in which both thermomechanical fields are fully coupled, we will however assume that the evolution is quasi-static and will hence neglect inertial effects. Note that a staggered approach could also have been adopted in which one field is calculated first say the temperature for instance using predicted values of the other field and similarly for the other field in a second step see for instance FAR Static elastic computation with thermal strains is treated in the Linear thermoelasticity weak coupling tour.

The problem consists of a quarter of a square plate perforated by a circular hole. Symmetry conditions are applied on the corresponding symmetry planes and stress and flux-free boundary conditions are adopted on the plate outer boundary. We first import the relevant modules and define the mesh and material parameters see the next section for more details on the parameters.

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We now define the relevant FunctionSpace for the considered problem. Since we will adopt a monolithic approach i. Let us just point out that the constructor using MixedFunctionSpace has been deprecated since version Dirichlet boundary conditions must be defined from the full FunctionSpace V using the appropriate subspaces that is V. These equations are completed by the equilibrium equation which will later be expressed in its weak form virtual work principle and the linearized heat equation without source terms :.

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Using the entropy constitutive relation, the weak form of the heat equation reads as:. For more details on the time discretization of the heat equation, see also the Heat equation FEniCS tutorial. These two forms are implemented below with zero right-hand sides zero Neumann BCs for both problems here. The problem is now solved by looping over time increments. Because of the typical exponential time variation of temperature evolution of the heat equation, time steps are discretized on a non-uniform logarithmic scale.

This evolution is plotted below. The final stresses and temperature variation are plotted using matplotlib. Note that we can add colorbar and change the plot axis limit. An unconditionally stable staggered algorithm for transient finite element analysis of coupled thermoelastic problems.

Computer Methods in Applied Mechanics and Engineering, 85 3 Numerical tours of continuum mechanics using FEniCS.This document presents a FEniCS tutorial to get new users quickly up and running with solving differential equations. The tutorial is still in an initial state so the reader is encouraged to send email to the author on hpl simula.

The goal of this tutorial is to get you started with FEniCS through a series of simple examples that demonstrate. The mathematics of the illustrations is kept simple to better focus on FEniCS functionality and syntax. This means that we mostly use the Poisson equation and the time-dependent diffusion equation as model problems, often with input data adjusted such that we get a very simple solution that can be exactly reproduced by any standard finite element method over a uniform, structured mesh.

This latter property greatly simplifies the verification of the implementations. Occasionally we insert a physically more relevant example to remind the reader that changing the PDE and boundary conditions to something more real might often be a trivial task. FEniCS may seem to require a thorough understanding of the abstract mathematical version of the finite element method as well as familiarity with the Python programming language.

Nevertheless, it turns out that many are able to pick up the fundamentals of finite elements and Python programming as they go along with this tutorial. Simply keep on reading and try out the examples.

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Reading this tutorial obviously requires access to a machine where the FEniCS software is installed. All the examples discussed in the following are available as executable Python source code files in a directory tree. Here, is the unknown function, is a prescribed function, is the Laplace operator also often written asis the spatial domain, and is the boundary of. A stationary PDE like this, together with a complete set of boundary conditions, constitute a boundary-value problemwhich must be precisely stated before it makes sense to start solving it with FEniCS.

In two space dimensions with coordinates andwe can write out the Poisson equation as. The unknown is now a function of two variables,defined over a two-dimensional domain. The Poisson equation arises in numerous physical contexts, including heat conduction, electrostatics, diffusion of substances, twisting of elastic rods, inviscid fluid flow, and water waves. Moreover, the equation appears in numerical splitting strategies of more complicated systems of PDEs, in particular the Navier-Stokes equations.

We shall now go through steps 2—4 in detail.

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The key feature of FEniCS is that steps 3 and 4 result in fairly short code, while most other software frameworks for PDEs require much more code and more technically difficult programming. FEniCS makes it easy to solve PDEs if finite elements are used for discretization in space and the problem is expressed as a variational problem.

Readers who are not familiar with variational problems will get a brief introduction to the topic in this tutorial, but getting and reading a proper book on the finite element method in addition is encouraged.Simple and extensible plate and shell finite element models through automatic code generation tools, J.

Hale, M. Brunetti, S. Bordas, C. However, you can use any method you want to install FEniCS. If you do not have an appropiate version of FEniCS already installed, use a Docker container skip the second line if you have already an appropiate version of FEniCS installed :.

You should now have a shell inside a container with FEniCS installed. Try out an example:. These files can be opened using Paraview.

We use Bitbucket Pipelines to perform automated testing. All documented demos include basic sanity checks on the results.

Tests are run in the quay.

FEniCS Project

We are always looking for contributions and help with fenics-shells. If you have ideas, nice applications or code contributions then we would be happy to help you get them included. Please use the bugtracker to report any issues. For support or questions please email jack. FEniCS-Shells latest. Contents API reference Documented demos. FEniCS-Shells is compatible with the FEniCS-Shells is described fully in the paper: Simple and extensible plate and shell finite element models through automatic code generation tools, J.

Additionally, the following models are under active development: linear and non-linear Timoshenko beams, We are using a variety of finite element numerical techniques including: MITC reduction operators, discontinuous Galerkin methods, reduced integration techniques.Task 1.

Discretize the equation in time and write variational formulation of the problem. Task 2. You can follow the procedure from subdomains-poisson demo. Follow a construction of boundaries object therein.

Task 3. According to your personal taste, get hint at Expression class documentation or any documented demo. Task 4. Task 5. Now proceed to variational formulation and time-stepping loop. Write bilinear and linear form representing PDE. How is solution at previous time-step represented therein? Hint See Poisson demo. Very short introduction to FEniCS. Stokes equation. Enter search terms or a module, class or function name.

Navigation index next previous FEniCS tutorial 1. Use LinearVariationalProblem and LinearVariationalSolver classes so that solve method of an instance of the latter is called every time-step while nothing else is touched excepted updating value of solution from previous time-step figuring in variational form. You can use for instance Function. Last updated on Feb 19, The goal of this tutorial is to get you started with FEniCS through a series of simple examples that demonstrate. The mathematics of the illustrations is kept simple to better focus on FEniCS functionality and syntax. This means that we mostly use the Poisson equation and the time-dependent diffusion equation as model problems, often with input data adjusted such that we get a very simple solution that can be exactly reproduced by any standard finite element method over a uniform, structured mesh.

This latter property greatly simplifies the verification of the implementations.

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Occasionally we insert a physically more relevant example to remind the reader that changing the PDE and boundary conditions to something more real might often be a trivial task. FEniCS may seem to require a thorough understanding of the abstract mathematical version of the finite element method as well as familiarity with the Python programming language. Nevertheless, it turns out that many are able to pick up the fundamentals of finite elements and Python programming as they go along with this tutorial.

A topology optimisation problem constrained by the Stokes equations

Simply keep on reading and try out the examples. Reading this tutorial obviously requires access to a machine where the FEniCS software is installed. All the examples discussed in the following are available as executable Python source code files in a directory tree. Here, is the unknown function, is a prescribed function, is the Laplace operator also often written asis the spatial domain, and is the boundary of.

A stationary PDE like this, together with a complete set of boundary conditions, constitute a boundary-value problemwhich must be precisely stated before it makes sense to start solving it with FEniCS. In two space dimensions with coordinates andwe can write out the Poisson equation as. The unknown is now a function of two variables,defined over a two-dimensional domain. The Poisson equation arises in numerous physical contexts, including heat conduction, electrostatics, diffusion of substances, twisting of elastic rods, inviscid fluid flow, and water waves. Moreover, the equation appears in numerical splitting strategies of more complicated systems of PDEs, in particular the Navier-Stokes equations.

We shall now go through steps 2—4 in detail. The key feature of FEniCS is that steps 3 and 4 result in fairly short code, while most other software frameworks for PDEs require much more code and more technically difficult programming. FEniCS makes it easy to solve PDEs if finite elements are used for discretization in space and the problem is expressed as a variational problem.

Readers who are not familiar with variational problems will get a brief introduction to the topic in this tutorial, but getting and reading a proper book on the finite element method in addition is encouraged.

The section Books on the Finite Element Method contains a list of some suitable books. The core of the recipe for turning a PDE into a variational problem is to multiply the PDE by a functionintegrate the resulting equation overand perform integration by parts of terms with second-order derivatives. The function which multiplies the PDE is in the mathematical finite element literature called a test function.

Documentation

The unknown function to be approximated is referred to as a trial function. The terms test and trial function are used in FEniCS programs too.This guide summarises how to install FEniCS. For installation in high performance computing clusters we recommend always building from source. FEniCS needs Python 3. For building CMake is needed and pip is recommended. To install it:. This will install FFC and its dependencies. Update DOLFIN documentation for latest release, including updates for building Python interface with pybind11, and provide a link to documentation of stable version here.

Consolidate multiple documentation locations: RTDwebserver. To download and build current development version run the following commands:. To get started, install Dockerand then run. A helper script is also available. For more options and features, see fenicsproject help. To install the latest FEniCS release from using conda :. FEniCS Project latest.

Todo Update DOLFIN documentation for latest release, including updates for building Python interface with pybind11, and provide a link to documentation of stable version here. Note Conda support is experimental and subject to changes.This program performs a dynamic modal analysis of an elastic cantilever beam represented by a 3D solid continuum.

The eigenmodes are computed using the SLEPcEigensolver and compared against an analytical solution of beam theory. The first two fundamental modes are on top with bending along the weak axis left and along the strong axis rightthe next two modes are at the bottom. The problem type is specified to be a generalized eigenvalue problem with Hermitian matrices.

By default, SLEPc computes the largest eigenvalues. Here we instead look for the smallest eigenvalues they should all be real. A spectral transform is therefore performed using the keyword shift-invert i.

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Numerical tours of continuum mechanics using FEniCS. The first four eigenmodes of this demo will look as follows: The first two fundamental modes are on top with bending along the weak axis left and along the strong axis rightthe next two modes are at the bottom. 