Difference between revisions of "Bim package"

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<pre>
 
<pre>
GammaD = bim2c_unknowns_on_side (mesh, [1 2]);           ## DIRICHLET NODES LIST
+
GammaD = bim2c_unknowns_on_side (mesh, [1 2]);     ## DIRICHLET NODES LIST
GammaN = bim2c_unknowns_on_side (mesh, [3 4]);           ## NEUMANN NODES LIST
+
GammaN = bim2c_unknowns_on_side (mesh, [3 4]);     ## NEUMANN NODES LIST
 
GammaN = setdiff (GammaN, GammaD);
 
GammaN = setdiff (GammaN, GammaD);
  
jn    = zeros (length (GammaN),1);                     ## PRESCRIBED NEUMANN FLUXES
+
jn    = zeros (length (GammaN),1);               ## PRESCRIBED NEUMANN FLUXES
ud    = 3*xu;                                             ## DIRICHLET DATUM
+
ud    = 3*xu;                                       ## DIRICHLET DATUM
 
Omega = setdiff (1:nnodes, union (GammaD, GammaN)); ## INTERIOR NODES LIST
 
Omega = setdiff (1:nnodes, union (GammaD, GammaN)); ## INTERIOR NODES LIST
 
</pre>
 
</pre>
Line 125: Line 125:
 
Ani = A(GammaN, Omega);  
 
Ani = A(GammaN, Omega);  
  
Aid = A(Ilist, GammaD);
+
Aid = A(Omega, GammaD);
Ain = A(Ilist, GammaN);  
+
Ain = A(Omega, GammaN);  
Aii = A(Ilist, Omega);  
+
Aii = A(Omega, Omega);  
  
 
bd = b(GammaD);
 
bd = b(GammaD);
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<pre>
 
<pre>
temp = [Ann Ani ; Ain Aii ] \ [ Fn+bn-And*ud(GammaD) ; bi-Aid*ud(GammaD)];
+
temp = [Ann Ani ; Ain Aii ] \ [ jn+bn-And*ud(GammaD) ; bi-Aid*ud(GammaD)];
u(Nlist)   = temp(1:numel (GammaN));
+
u = ud;
u(Omega)  = temp(length(un)+1:end);
+
u(GammaN) = temp(1:numel (GammaN));
 +
u(Omega)  = temp(length(GammaN)+1:end);
 
</pre>
 
</pre>
  
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<pre>
 
<pre>
Fd = [Add Adi Adn] * u([GammaD; Omega; GammaN]) - bd;
+
jd = [Add Adi Adn] * u([GammaD; Omega; GammaN]) - bd;
 
</pre>
 
</pre>
  
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<pre>
 
<pre>
[jxglob, jyglob] = bim2c_global_flux (mesh, u, epsilon, 1, phi);
+
[jxglob, jyglob] = bim2c_global_flux (mesh, u, epsilon*ones(nelems, 1), ones(nnodes, 1), ones(nnodes, 1), phi);
 
</pre>
 
</pre>
  
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or [[https://wci.llnl.gov/codes/visit/|visit]]
 
or [[https://wci.llnl.gov/codes/visit/|visit]]
 
<pre>
 
<pre>
fpl_vtk_write_field ("vtkdata", mesh, {}, {[gx; gy]', "Gradient"}, 1);
+
fpl_vtk_write_field ("vtkdata", mesh, {u, "Solution"}, {[gx; gy]', "Gradient"}, 1);
 
</pre>
 
</pre>
  
 
[[Category:OctaveForge]]
 
[[Category:OctaveForge]]
 
[[Category:Packages]]
 
[[Category:Packages]]

Revision as of 15:38, 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


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, 1, phi);
Mass    = bim2a_reaction (mesh, 1, 1);
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 we let be composed by sides 1 and 2 and be the rest of the boundary

GammaD = bim2c_unknowns_on_side (mesh, [1 2]); 	    ## DIRICHLET NODES LIST
GammaN = bim2c_unknowns_on_side (mesh, [3 4]); 	    ## NEUMANN NODES LIST
GammaN = setdiff (GammaN, GammaD);

jn    = zeros (length (GammaN),1);           	    ## PRESCRIBED NEUMANN FLUXES
ud    = 3*xu;                                       ## DIRICHLET DATUM
Omega = setdiff (1:nnodes, union (GammaD, GammaN)); ## INTERIOR NODES LIST


Add = A(GammaD, GammaD);
Adn = A(GammaD, GammaN); ## shoud be all zeros hopefully!!
Adi = A(GammaD, Omega); 

And = A(GammaN, GammaD); ## shoud be all zeros hopefully!!
Ann = A(GammaN, GammaN);
Ani = A(GammaN, Omega); 

Aid = A(Omega, GammaD);
Ain = A(Omega, GammaN); 
Aii = A(Omega, Omega); 

bd = b(GammaD);
bn = b(GammaN); 
bi = b(Omega); 

Solve for the displacements

temp = [Ann Ani ; Ain Aii ] \ [ jn+bn-And*ud(GammaD) ; bi-Aid*ud(GammaD)];
u = ud;
u(GammaN)  = temp(1:numel (GammaN));
u(Omega)   = temp(length(GammaN)+1:end);

Compute the fluxes through Dirichlet sides

jd = [Add Adi Adn] * u([GammaD; Omega; GammaN]) - 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, epsilon*ones(nelems, 1), ones(nnodes, 1), ones(nnodes, 1), phi);

Export data to VTK format

The resut can be exported to vtk format to visualize with [[1]] or [[2]]

fpl_vtk_write_field ("vtkdata", mesh, {u, "Solution"}, {[gx; gy]', "Gradient"}, 1);