The FEniCS computing platform

The FEniCS computing platform

FEniCS is a popular computing platform for partial differential equations (PDE). FEniCS enables users to quickly translate scientific models into efficient finite element code. With the high-level Python and C++ interfaces to FEniCS, it is easy to get started, but FEniCS offers also powerful capabilities for more experienced programmers. FEniCS runs on a multitude of platforms ranging from laptops to high-performance clusters.

Solving a PDE in FEniCS

As an illustration of how to program a simple PDE model with FEniCS, consider the Stokes equations in variational form:

∫Ωgradu:gradvdx–∫Ωpdivvdx+∫Ωdivuqdx=∫Ωf⋅vdx.

The variational problem is easily transcribed into Python using mathematical operators in FEniCS:

# Define function space P2 = VectorElement('P', tetrahedron, 2) P1 = FiniteElement('P', tetrahedron, 1) TH = P2 * P1 W = FunctionSpace(mesh, TH) # Define variational problem (u, p) = TrialFunctions(W) (v, q) = TestFunctions(W) a = inner(grad(u), grad(v))*dx - p*div(v)*dx + div(u)*q*dx L = dot(f, v)*dx # Compute solution w = Function(W) solve(a == L, w, [bc1, bc0])

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# Define function space P2 = VectorElement ( 'P' , tetrahedron , 2 ) P1 = FiniteElement ( 'P' , tetrahedron , 1 ) TH = P2 * P1 W = FunctionSpace ( mesh , TH )   # Define variational problem ( u , p ) = TrialFunctions ( W ) ( v , q ) = TestFunctions ( W ) a = inner ( grad ( u ) , grad ( v ) ) * dx - p * div ( v ) * dx + div ( u ) * q * dx L = dot ( f , v ) * dx   # Compute solution w = Function ( W ) solve ( a == L , w , [ bc1 , bc0 ] )

The above code snippet also shows how to define a suitable finite element function space, using continuous piecewise quadratic vector-valued functions for the velocity and continuous piecewise linear functions for the pressure (Taylor-Hood). The computational domain and mesh are also easily created with FEniCS, here defined by three spheres immersed in a 3D channel.

# Define domain h = 0.25 r = 0.3*h box = Box(Point(0, 0, 0), Point(1, h, h)) s0 = Sphere(Point(0.3, 0.50*h, 0.50*h), r) s1 = Sphere(Point(0.5, 0.65*h, 0.65*h), r) s2 = Sphere(Point(0.7, 0.35*h, 0.35*h), r) domain = box - s0 - s1 - s2 # Generate mesh mesh = generate_mesh(domain, 32)

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# Define domain h = 0.25 r = 0.3 * h box = Box ( Point ( 0 , 0 , 0 ) , Point ( 1 , h , h ) ) s0 = Sphere ( Point ( 0.3 , 0.50 * h , 0.50 * h ) , r ) s1 = Sphere ( Point ( 0.5 , 0.65 * h , 0.65 * h ) , r ) s2 = Sphere ( Point ( 0.7 , 0.35 * h , 0.35 * h ) , r ) domain = box - s0 - s1 - s2   # Generate mesh mesh = generate_mesh ( domain , 32 )

High-performance computing

 

Each component of the FEniCS platform has been fundamentally designed for parallel processing. Executing a FEniCS script in parallel is as simple as callingmpirun -np 8 python script.py. This framework allows for rapid prototyping of finite element formulations and solvers on laptops and workstations, and the same code may then be deployed on large high-performance computers.

The figure shows the thermal component of a nonlinear thermomechanical FEniCS simulation of a steam turbine. The finite element system of linear equations comprises more than 10⁸ degrees of freedom. The solver was initially developed on a desktop computer for a small scale problem. The same code was then deployed on a supercomputer with 512 parallel processes distributed over 32 nodes.

Installation and documentation

FEniCS is available for a range of platforms (Linux, Mac, Windows). Choose between Docker containers, binary packages and source code. Visit ourinstallation page to get the latest version of FEniCS. FEniCS comes withextensive documentation and numerous examples. A good starting point is theFEniCS Tutorial.

About

The FEniCS Project is developed and maintained as a freely available, open-source project by a global community of scientists and software developers. The project is developed by the FEniCS Community, isgoverned by theFEniCS Steering Council and is overseen by the FEniCS Advisory Board.

FEniCS is a NumFOCUS fiscally supported project. If you like FEniCS and want to support our mission to produce the best possible platform for open-source computing, consider making adonation to our project.

 

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FEniCS‘17

Download

The latest stable release of FEniCS is version 2016.2.0, which was released on December 6 2016. Choose between Docker containers, binary packages and source code.

Documentation

Our documentation includes several books, a collection of documented demo programs (notebooks), and a reference manual.

Community

Get in touch with the FEniCS community, discuss your issues and share your knowledge with other users, and hook up with developers to contribute to the future development of FEniCS.