Difference between revisions of "Bim package"
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This is a short example on how to use <tt>bim</tt> to solve a 2D Diffusion Advection Reaction problem.  This is a short example on how to use <tt>bim</tt> to solve a 2D Diffusion Advection Reaction problem.  
−  The  +  The complete code for this example is on [[Agora]] at this [http://agora.octave.org/snippet/1bqV link]. 
We want to solve the equation  We want to solve the equation 
Revision as of 07:31, 24 July 2012
Package for solving Diffusion Advection Reaction (DAR) Partial Differential Equations based on the Finite Volume ScharfetterGummel (FVSG) method a.k.a Box Integration Method (BIM).
Contents
Tutorials
2D Diffusion Advection Reaction example
This is a short example on how to use bim to solve a 2D Diffusion Advection Reaction problem. The complete code for this example is on Agora at this link.
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
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
Code: Meshing the 2D problem 
[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
Code: Visualizing the mesh for the 2D problem 
pdemesh (mesh.p, mesh.e, mesh.t)
view (2)

Set the coefficients for the problem:
Get the node coordinates from the mesh structure
Code: Get mesh coordinates in the 2D problem 
xu = mesh.p(1,:).';
yu = mesh.p(2,:).';

Get the number of elements and nodes in the mesh
Code: Get number of elements in the 2D problem 
nelems = columns (mesh.t);
nnodes = columns (mesh.p);

Code: Set value of coefficients for the 2D problem 
epsilon = .1;
phi = xu + yu;

Construct the discretized operators
Code: Discretized operators for the 2D problem 
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
Code: Boundary conditions for the 2D problem 
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 tracer density
Code: Compute solution of the 2D problem 
temp = [Ann Ani ; Ain Aii ] \ [ jn+bnAnd*ud(GammaD) ; biAid*ud(GammaD)];
u = ud;
u(GammaN) = temp(1:numel (GammaN));
u(Omega) = temp(length(GammaN)+1:end);

Compute the fluxes through Dirichlet sides
Code: Boundary fluxes in the 2D problem 
jd = [Add Adi Adn] * u([GammaD; Omega; GammaN])  bd;

Compute the gradient of the solution
Code: Gradient of solution in the 2D problem 
[gx, gy] = bim2c_pde_gradient (mesh, u);

Compute the internal AdvectionDiffusion flux
Code: Total flux for the 2D problem 
[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]]
Code: Export the solution of the 2D problem to vtk 
fpl_vtk_write_field ("vtkdata", mesh, {u, "Solution"}, {[gx; gy]', "Gradient"}, 1);

you can also plot your data directly in Octave using pdesurf
Code: Rubbersheet visualization of the solution of the 2D problem 
pdesurf (mesh.p, mesh.t, u)

it will look like this
3D Time dependent problem
Here is an example of a 3D timedependent AdvectionDiffusion equation that uses lsode
for timestepping.
The equation being solved is
The initial condition is
Code: Define the 3D problem 
pkg load bim
x = linspace (0, 1, 40);
y = z = linspace (0, 1, 20);
msh = bim3c_mesh_properties (msh3m_structured_mesh (x, y, z, 1, 1:6));
nn = columns (msh.p);
ne = columns (msh.t);
x = msh.p(1, :).';
y = msh.p(2, :).';
z = msh.p(3, :).';
x0 = .2; sx = .1;
y0 = .2; sy = .1;
z0 = .8; sz = .1;
u = exp ( ((xx0)/(2*sx)) .^2  ((yy0)/(2*sy)) .^2  ((zz0)/(2*sz)) .^2);
A = bim3a_advection_diffusion (msh, .01*ones(ne, 1), 100*(x+yz));
M = bim3a_reaction (msh, 1, 1);
function du = f (u, t, A, M)
du =  M \ (A * u);
endfunction
time = linspace (0, 1, 30);
lsode_options ("integration method", "adams");
U = lsode (@(u, t) f(u, t, A, M), u, time);
for ii = 1:1:numel (time)
name = sprintf ("u_%3.3d", ii);
delete ([name ".vtu"]);
fpl_vtk_write_field (name, msh, {U(ii,:)', 'u'}, {}, 1);
endfor

This is a video showing the .3 isosurface of the solution.