Difference between revisions of "Bim package"

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<math> u(x, y) = u_d(x, y)\qquad \mbox{ on } \Gamma_D </math>
 
<math> u(x, y) = u_d(x, y)\qquad \mbox{ on } \Gamma_D </math>
  
<math> -( \varepsilon\ \nabla u(x, y) - \nabla \varphi(x,y)\ u(x, y) )  \cdot \mathbf{n} \qquad \mbox{ on } \Gamma_N</math>
+
<math> -( \varepsilon\ \nabla u(x, y) - \nabla \varphi(x,y)\ u(x, y) )  \cdot \mathbf{n} = j_N(x, y)\qquad \mbox{ on } \Gamma_N</math>
  
 
<b> Create the mesh and precompute the mesh properties </b>
 
<b> Create the mesh and precompute the mesh properties </b>
  
The geometry of the domain was created using gmsh and is stored in the file <tt>fiume.geo</tt>
+
To define the geometry of the domain we can use [http://gmsh.geuz.org gmsh].
created with [http://gmsh.geuz.org gmsh]
+
 
 +
the following gmsh input
 +
 
 +
<pre>
 +
Point (1)  = {0, 0, 0, 0.1};
 +
Point (2)  = {1, 1, 0, 0.1};
 +
Point (3)  = {1, 0.9, 0, 0.1};
 +
Point (4)  = {0, 0.1, 0, 0.1};
 +
Point (5) = {0.3,0.1,-0,0.1};
 +
Point (6) = {0.4,0.4,-0,0.1};
 +
Point (7) = {0.5,0.6,0,0.1};
 +
Point (8) = {0.6,0.9,0,0.1};
 +
Point (9) = {0.8,0.8,0,0.1};
 +
Point (10) = {0.2,0.2,-0,0.1};
 +
Point (11) = {0.3,0.5,0,0.1};
 +
Point (12) = {0.4,0.7,0,0.1};
 +
Point (13) = {0.5,1,0,0.1};
 +
Point (14) = {0.8,0.9,0,0.1};
 +
 
 +
Line (1)  = {3, 2};
 +
Line (2) = {4, 1};
 +
 
 +
CatmullRom(3) = {1,5,6,7,8,9,3};
 +
CatmullRom(4) = {4,10,11,12,13,14,2};
 +
Line Loop(15) = {3,1,-4,2};
 +
Plane Surface(16) = {15};
 +
</pre>
 +
 
 +
will produce the geometry below
  
 
[[File:fiume.png]]
 
[[File:fiume.png]]
 +
 +
we need to load the mesh into Octave and precompute mesh properties
 +
check out the tutorial for the [[msh_package|msh package]] for info
 +
on the mesh structure
  
 
<pre>
 
<pre>
Line 27: Line 59:
 
</pre>
 
</pre>
  
to see the mesh you can use functions from the [fpl] package
+
to see the mesh you can use functions from the [[fpl_package|fpl package]]
  
 
<pre>
 
<pre>
Line 38: Line 70:
 
<b> Construct an initial guess</b>
 
<b> Construct an initial guess</b>
  
We need this even if our problem is linear and stationary
+
<b> Set the coefficients for the problem:</b>
as we are going to use the values at boundary nodes to set
 
Dirichelet boundary conditions.
 
  
 
Get the node coordinates from the mesh structure
 
Get the node coordinates from the mesh structure
Line 49: Line 79:
 
</pre>
 
</pre>
  
 
<pre>
 
uin    = 3*xu;
 
</pre>
 
 
<b> Set the coefficients for the problem:</b>
 
  
 
Get the number of elements and nodes in the mesh
 
Get the number of elements and nodes in the mesh
Line 65: Line 89:
 
<pre>
 
<pre>
 
epsilon = .1;
 
epsilon = .1;
phi    = xu+yu;
+
phi    = xu + yu;
 
</pre>
 
</pre>
  
Line 82: Line 106:
  
 
<pre>
 
<pre>
Dlist = bim2c_unknowns_on_side(mesh, [8 18]);   ## DIRICHLET NODES LIST
+
GammaD = bim2c_unknowns_on_side(mesh, [1 2]);   ## DIRICHLET NODES LIST
Nlist = bim2c_unknowns_on_side(mesh, [23 24]);   ## NEUMANN NODES LIST
+
GammaN = bim2c_unknowns_on_side(mesh, [3 4]);   ## NEUMANN NODES LIST
Nlist = setdiff(Nlist,Dlist);
+
Corners = setdiff(GammaD,GammaN);
Fn   = zeros(length(Nlist),1);            ## PRESCRIBED NEUMANN FLUXES
+
jn   = zeros(length(GammaN),1);            ## PRESCRIBED NEUMANN FLUXES
 +
ud    = 3*xu;                                      ## DIRICHLET DATUM
 
Ilist = setdiff(1:length(uin),union(Dlist,Nlist)); ## INTERNAL NODES LIST
 
Ilist = setdiff(1:length(uin),union(Dlist,Nlist)); ## INTERNAL NODES LIST
 
</pre>
 
</pre>

Revision as of 14:51, 19 July 2012

This is a short example on how to use bim to solve a DAR problem.
The data for this example can be found in the doc directory inside the bim installation directory.

We want to solve the equation

with mixed Dirichlet / Neumann boundary conditions

Create the mesh and precompute the mesh properties

To define the geometry of the domain we can use gmsh.

the following gmsh input

Point (1)  = {0, 0, 0, 0.1};
Point (2)  = {1, 1, 0, 0.1};
Point (3)  = {1, 0.9, 0, 0.1};
Point (4)  = {0, 0.1, 0, 0.1};
Point (5) = {0.3,0.1,-0,0.1};
Point (6) = {0.4,0.4,-0,0.1};
Point (7) = {0.5,0.6,0,0.1};
Point (8) = {0.6,0.9,0,0.1};
Point (9) = {0.8,0.8,0,0.1};
Point (10) = {0.2,0.2,-0,0.1};
Point (11) = {0.3,0.5,0,0.1};
Point (12) = {0.4,0.7,0,0.1};
Point (13) = {0.5,1,0,0.1};
Point (14) = {0.8,0.9,0,0.1};

Line (1)  = {3, 2};
Line (2) = {4, 1};

CatmullRom(3) = {1,5,6,7,8,9,3};
CatmullRom(4) = {4,10,11,12,13,14,2};
Line Loop(15) = {3,1,-4,2};
Plane Surface(16) = {15};

will produce the geometry below

Fiume.png

we need to load the mesh into Octave and precompute mesh properties check out the tutorial for the msh package for info on the mesh structure

[mesh] = msh2m_gmsh("fiume","scale",1,"clscale",.1);
[mesh] = bim2c_mesh_properties(mesh);

to see the mesh you can use functions from the fpl package

pdemesh (mesh.p, mesh.e, mesh.t)
view (2)

Fiume msh.png

Construct an initial guess

Set the coefficients for the problem:

Get the node coordinates from the mesh structure

xu     = mesh.p(1,:).';
yu     = mesh.p(2,:).';


Get the number of elements and nodes in the mesh

nelems = columns(mesh.t);
nnodes = columns(mesh.p);
epsilon = .1;
phi     = xu + yu;

Construct the discretized operators

AdvDiff = bim2a_advection_diffusion(mesh, epsilon, 1, phi);
Mass    = bim2a_reaction(mesh,delta,zeta);
b       = bim2a_rhs(mesh,f,g);
A       = AdvDiff + Mass;

To Apply Boundary Conditions, partition LHS and RHS

The tags of the sides are assigned by gmsh

GammaD = bim2c_unknowns_on_side(mesh, [1 2]); 	   ## DIRICHLET NODES LIST
GammaN = bim2c_unknowns_on_side(mesh, [3 4]); 	   ## NEUMANN NODES LIST
Corners = setdiff(GammaD,GammaN);
jn    = zeros(length(GammaN),1);           	   ## PRESCRIBED NEUMANN FLUXES
ud    = 3*xu;                                      ## DIRICHLET DATUM
Ilist = setdiff(1:length(uin),union(Dlist,Nlist)); ## INTERNAL NODES LIST


Add = A(Dlist,Dlist);
Adn = A(Dlist,Nlist); ## shoud be all zeros hopefully!!
Adi = A(Dlist,Ilist); 

And = A(Nlist,Dlist); ## shoud be all zeros hopefully!!
Ann = A(Nlist,Nlist);
Ani = A(Nlist,Ilist); 

Aid = A(Ilist,Dlist);
Ain = A(Ilist,Nlist); 
Aii = A(Ilist,Ilist); 

bd = b(Dlist);
bn = b(Nlist); 
bi = b(Ilist); 

ud = uin(Dlist);
un = uin(Nlist); 
ui = uin(Ilist); 

Solve for the displacements

temp = [Ann Ani ; Ain Aii ] \ [ Fn+bn-And*ud ; bi-Aid*ud];
un   = temp(1:length(un));
ui   = temp(length(un)+1:end);
u(Dlist) = ud;
u(Ilist) = ui;
u(Nlist) = un;

Compute the fluxes through Dirichlet sides

Fd = Add * ud + Adi * ui + Adn*un - bd;


Compute the gradient of the solution

[gx, gy] = bim2c_pde_gradient(mesh,u);

Compute the internal Advection-Diffusion flux

[jxglob,jyglob] = bim2c_global_flux(mesh,u,alfa,gamma,eta,beta);

Save data for later visualization

fpl_dx_write_field("dxdata",mesh,[gx; gy]',"Gradient",1,2,1);
fpl_vtk_write_field ("vtkdata", mesh, {}, {[gx; gy]', "Gradient"}, 1);