short introduction to freefem

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A short introduction to FreeFem++

M. Ersoy

Basque Center for Applied Mathematics

12 November 2010


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Outline of the talkOutline of the talk

1 Introduction

2 HOW TOsolve steady pdes : e.g. Laplace equationsolve unsteady pdes : e.g. Heat equation

M. Ersoy (BCAM) Freefem++ 12 November 2010 2 / 22


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OutlineOutline

1 Introduction




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A software for solving PDEs

FreeFem++ for 2D-3D 1 PDEsFinite element method

Freesoftware : available at http://www.freefem.org/ff++/ 2 , it runs on Linux Windows

Mac

1. in progress2. Useful documentation available at http://www.freefem.org/ff++/ftp/freefem++doc.

pdfM. Ersoy (BCAM) Freefem++ 12 November 2010 4 / 22
http://www.freefem.org/ff++/http://www.freefem.org/ff++/ftp/freefem++doc.pdfhttp://www.freefem.org/ff++/ftp/freefem++doc.pdfhttp://www.freefem.org/ff++/ftp/freefem++doc.pdfhttp://www.freefem.org/ff++/ftp/freefem++doc.pdfhttp://www.freefem.org/ff++/ftp/HISTORYhttp://www.freefem.org/ff++/


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FreeFem++for 2D-3D1

PDEsFinite element method

Free software : available at http://www.freefem.org/ff++/ 2 , it runs on Linux Windows

Mac++extension of FreeFem and FreeFem+ (see Historichttp://www.freefem.org/ff++/ftp/HISTORY)


http://www.freefem.org/ff++/http://www.freefem.org/ff++/ftp/HISTORYhttp://www.freefem.org/ff++/ftp/freefem++doc.pdfhttp://www.freefem.org/ff++/ftp/freefem++doc.pdfhttp://www.freefem.org/ff++/ftp/freefem++doc.pdfhttp://www.freefem.org/ff++/ftp/freefem++doc.pdfhttp://www.freefem.org/ff++/ftp/HISTORYhttp://www.freefem.org/ff++/


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FreeFem++ for 2D-3D1

PDEsFinite element method

Free software : available at http://www.freefem.org/ff++/ 2 , it runs on Linux Windows

Mac++ extension of FreeFem and FreeFem+ (see Historichttp://www.freefem.org/ff++/ftp/HISTORY)

FreeFem++ team : Olivier Pironneau, Frederic Hecht, Antoine Le Hyaric,Jacques Morice


http://www.freefem.org/ff++/http://www.freefem.org/ff++/ftp/HISTORYhttp://www.freefem.org/ff++/ftp/freefem++doc.pdfhttp://www.freefem.org/ff++/ftp/freefem++doc.pdfhttp://www.freefem.org/ff++/ftp/freefem++doc.pdfhttp://www.freefem.org/ff++/ftp/freefem++doc.pdfhttp://www.freefem.org/ff++/ftp/HISTORYhttp://www.freefem.org/ff++/


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OutlineOutline

1 Introduction




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OutlineOutline

1 Introduction




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The problem

Given fL2(), find H10 () such that :

= f in (x1, x2) = 0 on 1

n:= n= 0 on 2



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Variational formulation (VF)

Let w be a test function, the VF of the Laplace equation is :

w dx=

f w d x

or equivalentlyA(, w) =l(w)

where the bilinear form is

A(, w) =

w dx

and the linear one :l(w) =

f w d x



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Variational problem

Problem is now to find vH10 () such that, for every wH10 () :

A(, w) =l(w)

where the bilinear form is

A(, w) =

w dx

and the linear one :

l(w) = f w d xWe set V =H1() in the next



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FEM : Galerkin method

Let Vh V with dimVh=nh,

Vh =

u(x, y); u(x, y) =

nhk=1

ukk(x, y), uk R

where k Ps is a polynom of degree s.Now, the problem is to find hVh such that :

wVh, A(h, w) =l(w)

So, we have to solve :

i {1, . . . , nh}, A(j , i) =l(i)



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Spaces VhSpaces Vh =Vh(h, P) will depend on the mesh :

h=

n

k=1

Tk

andthe approximation PP0 piecewise constant approximationP1 C

0 piecewise linear approximation

...M. Ersoy (BCAM) Freefem++ 12 November 2010 11 / 22


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With FreeFem++

To numerically solve the Laplace equation, we have to :

1 mesh the domain (define the border + mesh)

2 write the VF3 show the result

its easy !



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Step 1 : mesh of the domain

if = 1+ 2+. . .thenFreeFem++ define border commands : borderGamma1(t = t0,tf){x = gam11(t), y = gam12(t)} ; borderGamma2(t = t0,tf){x = gam21(t), y = gam22(t)} ; ...

where the set {x = gam11(t), y = gam12(t)} is a parametrisation of theborder Gamma1 with t [t0, tf]

http://edp/maillage.edp


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Step 1 : mesh of the domain

if = 1+ 2+. . .thenFreeFem++ define border commands : borderGamma1(t = t0,tf){x = gam11(t), y = gam12(t)} ; borderGamma2(t = t0,tf){x = gam21(t), y = gam22(t)} ; ...

where the set {x = gam11(t), y = gam12(t)} is a parametrisation of theborder Gamma1 with t [t0, tf]

FreeFem++ mesh commands :

meshMeshName =buildmesh(Gamma1(m1)+Gamma1(m2)+. . . ) ;

where mi are positive (or negative) numbers to indicate the number of pointshould on j . example

http://edp/maillage.edphttp://edp/maillage.edp


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S 2 VF


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Step 2 : write the VFPremliminar

To write the VF, we need to

define finite element space , we will use :

fespaceNameFEspace(MeshName,P) ;

where P = P0 or P1 or P2, . . . . Example :fespaceVh(Omegah,P2) ;

use interpolated function, for instance,

Vh f = x+y ;

N.B. x and y are reserved key words


S 2 VF


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Step 2 : write the VFVF

VF is simply what we write on paper, for instance, for the Laplace equation, weshould have :

Vh phi, w, f ;

problemLaplace(phi,w)int2d(Omegah)(dx(phi)*dx(w) + dy(phi)*dy(w))- int2d(Omegah)(f*w)

+ boundary conditions;

where w, phi belong to FE space Vh.


S 2 VF


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Vh phi, w, f ;



where w, phi belong to FE space Vh.Next, to solve it, just write :

Laplace ;


S 2 VF


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Vh phi, w, f ;



where w, phi belong to FE space Vh.

or equivalently write directly :

solveLaplace(phi,w)int2d(Omegah)(dx(phi)*dx(w) + dy(phi)*dy(w))-int2d(Omegah)(f*w)+ boundary conditions

;M. Ersoy (BCAM) Freefem++ 12 November 2010 15 / 22

Step 2 write the VF


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Step 2 : write the VFBoundary conditions 3

Dirichlet condition u= g : +on(BorderName, u=g)

Neumann condition nu= g : -int1d(Th)( g*w). . .

3. see the manual p. 142M. Ersoy (BCAM) Freefem++ 12 November 2010 16 / 22

Step 3 : show the result


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Step 3 : show the result

to plot

plot(phi) ;

or

plot([dx(phi),dy(phi)]) ;

Laplace equation


Outline
http://edp/Laplace.edphttp://edp/Laplace.edp


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Outline

1 Introduction



The problem


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The problem

Given fL2(), find H1() such that :

t= f in (x1, x2) =z(x1, x2) on 1n:= n= 0 on 2

may be rewritten, after an implicit Euler finite difference approximation in time asfollows :

n+1 n

t n+1 =fn


Variational formulation


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Variational formulation

The Variational formulation for the semi discrete equation is : let w be a testfunction, we write :

n+1 nt

w+n+1 w dx=

fnw dx


The FreeFem formulation


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The FreeFem formulation

Following the steady state case, the problem is then :

solveLaplace(phi,w)int2d(Omegah)(phi*w/dt+dx(phi)*dx(w) + dy(phi)*dy(w))

-int2d(Omegah)(phiold*w/dtf*w)+ boundary conditions;

where we have to solve iteratively this discrete equation. example

http://edp/evo1.edphttp://edp/evo1.edp


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Enjoy

yourself


short introduction to freefem

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