Bim package

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Package for solving Diffusion Advection Reaction (DAR) Partial Differential Equations based on the Finite Volume Scharfetter-Gummel (FVSG) method a.k.a Box Integration Method (BIM).

Tutorials[edit]

2D Diffusion Advection Reaction example[edit]

This is a short example on how to use bim to solve a 2D Diffusion Advection Reaction problem. .

We want to solve the equation

  -\mathrm{div}\ ( \varepsilon\ \nabla u(x, y) - \nabla \varphi(x,y)\ u(x, y) ) ) + u(x, y) = 1 \qquad \mbox{ in } \Omega

  \varphi(x, y)\ =\ x + y

with mixed Dirichlet / Neumann boundary conditions

 u(x, y) = u_d(x, y)\qquad \mbox{ on } \Gamma_D

 -( \varepsilon\ \nabla u(x, y) - \nabla \varphi(x,y)\ u(x, y) )  \cdot \mathbf{n} = j_N(x, y)\qquad \mbox{ on } \Gamma_N

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

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)

Fiume msh.png


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  \Gamma_D be composed by sides 1 and 2 and  \Gamma_D 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+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

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 Advection-Diffusion 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

Fiume sol pdesurf.png

3D Time dependent problem[edit]

Here is an example of a 3D time-dependent Advection-Diffusion equation that uses lsode for time-stepping.

The equation being solved is

 \frac{\partial u}{\partial t} - \mathrm{div } \left(.01 \nabla u - u \nabla \varphi \right) 
= 0 \qquad \mbox{ in } \Omega \times [0, T] =[0, 1]^3 \times [0, 1]

 ~\varphi = x + y - z

 - \left(.01 \nabla u - u \nabla \varphi \right)  \cdot \mathbf{n} = 0 
\qquad \mbox{ on } \partial \Omega

The initial condition is

 u = \exp (- \left(\frac{x-.2}{.2}\right)^2 - \left(\frac{y-.2}{.2}\right)^2 - \left(\frac{z-.2}{.2}\right)^2)

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 (- ((x-x0)/(2*sx)) .^2 - ((y-y0)/(2*sy)) .^2 - ((z-z0)/(2*sz)) .^2);

A = bim3a_advection_diffusion (msh, .01*ones(ne, 1), 100*(x+y-z));
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.

External links[edit]