using fenics for finite element methods

Using FEniCS for Finite Element Methods

Andrew Gillette

UC San DiegoDepartment of Mathematics

August 2012

slides adapted from a longer presentation byJohan Hake, Simula Research Laboratory, Norway

Andrew Gillette - UCSD ()FEniCS for Finite Element Methods NBCR Summer Institute 2012 1 / 26

The finite element method (FEM) can be used to

discretize and solve PDEs for general geometries

∇2u = 0

u(x) = 100 at B1

u(x) = 0 at B2


FEniCS and PyDOLFIN are computational tools that

bundle a lot of subtle problem setup into a simple

interface

∂c

∂t= D∇2c

www.fenics.org


We will first use FEniCS to solve a stationary diffusion

problem, the Poisson equation on the unit interval

Poisson

−u′′ = f ; u′(0) = 4; u(1) = 0

Source, f(x) Solution, u(x)


The problem can be stated, solved and plotted, using

a PyDOLFIN script written in 16 lines

−u′′ = f

u′(0) = 4

u(1) = 0




−u′′ = f

u′(0) = 4

u(1) = 0


How did this become a discrete problem?

A FunctionSpace in PyDOLFIN takes a mesh and a finite element asarguments.

The first order Lagrange is the finite element described by thepiecewise linear nodal basis functions.


The PDE can be re-written using the discrete weak

formulation

Strong formulation

−u′′ = f

Should be true for every point (strong)in space

Discrete weak formulation

u(x) =5∑

j=0

ujφj(x)

−∫ 1

0u′′φidx =

∫ 1

0fφidx, i = 0, ...,5

By weighting theequation with φi andtaking the integralover the wholedomain, we solve anapproximation of u(weak)


Integrate the left side ‘by parts’ and simplify

−∫ 1

0u′′φidx =

∫ 1

0u′φ′idx + u′(0)φi(0)− u′(1)φi(1)

This includes the derivatives at the boundaries!

Recall that our boundary conditions implies thatφi(1) = 0 and u′(0) = 4, which gives us:∫ 1

0u′φ′idx =

∫ 1

0fφidx− 4φi(1), i = 0, ...,5

We use v instead of φi and write the variational problem:

Find u ∈ V such that

a(u, v) = L(v), ∀v ∈ V, where

a(u, v) =

∫Ω

u′v′dx and L(v) =

∫Ω

f v dx−∫∂Ω

4v ds


FEniCS can be used to describe variational forms,

and PyDOLFIN can be used to solve

VariationalProblems

Mathematical notation: PyDOLFIN notation:

Find u ∈ V such thata(u, v) = L(v), ∀v ∈ V

where:

a(u, v) =

∫Ω

u′v′dx

L(v) =

∫Ω

f v dx−∫∂Ω

4v ds


The approach applies to 2D and 3D

−∇2u = f ; ∂u∂n(:, [0,1]) = g; u([0,1], :) = 0


Time dependent system, like the diffusion equation,

can be solved using the same framework

PDE

u = D∇2u in Ω

D∂u∂n = J(t) on ∂Ω1

u = 1 on ∂Ω2

u(0, :) = 1

J(t) =

100 : t ≤ Jstop

0 : t > Jstop


Diffusion equation solved using PyDOLFIN

(Domain declarations)



(linear algebra initialization)


FEniCS is used in real, modern science!

Steady-state Ca2+ distributions about Troponin C, showing theabsence and presence of an electrostatic potential.

Kekenes-Huskey, Gillette, Hake, McCammonFinite Element Estimation of Protein-Ligand Association Rates with Post-Encounter Effects:

Applications to Calcium binding in Troponin C and SERCA,Submitted 2012.


Questions?

Slides from this presentation are available at my website:

http://ccom.ucsd.edu/∼agillette

Thanks to Johan Hake for preparing many of these slides!


using fenics for finite element methods

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