Editing Bim package

The {{Forge|bim}} package is part of the [[Octave Forge]] project.  It is a 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 ==
=== 2D Diffusion Advection Reaction example ===

This is a short example on how to use <tt>bim</tt> to solve a 2D Diffusion Advection Reaction problem.
<!-- The complete code for this example is on [[Agora]] at this [http://agora.octave.org/snippet/1bqV link] -->.

We want to solve the equation

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

<math>  \varphi(x, y)\ =\ x + y </math>

with mixed Dirichlet / Neumann boundary conditions

<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} = j_N(x, y)\qquad \mbox{ on } \Gamma_N</math>

<b> Create the mesh and precompute the mesh properties </b>

To define the geometry of the domain we can use [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]]

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

{{Code|Meshing the 2D problem|<syntaxhighlight lang="octave" style="font-size:13px">
[mesh] = msh2m_gmsh ("fiume","scale",1,"clscale",.1);
[mesh] = bim2c_mesh_properties (mesh);
</syntaxhighlight>}}

to see the mesh you can use functions from the [[fpl_package|fpl package]]

{{Code|Plot mesh in the 2D problem|<syntaxhighlight lang="octave" style="font-size:13px">
pdemesh (mesh.p, mesh.e, mesh.t)
view (2)
</syntaxhighlight>}} 

[[File:fiume_msh.png]]


<b> Set the coefficients for the problem:</b>

Get the node coordinates from the mesh structure

{{Code|Get mesh coordinates in the 2D problem|<syntaxhighlight lang="octave" style="font-size:13px">
xu     = mesh.p(1,:).';
yu     = mesh.p(2,:).';
</syntaxhighlight>}} 


Get the number of elements and nodes in the mesh

{{Code|Get number of elements in the 2D problem|<syntaxhighlight lang="octave" style="font-size:13px">
nelems = columns (mesh.t);
nnodes = columns (mesh.p);
</syntaxhighlight>}} 

<pre>
epsilon = .1;
phi     = xu + yu;
</pre>

<b> Construct the discretized operators</b>

{{Code|Discretized operators for the 2D problem|<syntaxhighlight lang="octave" style="font-size:13px"> 
AdvDiff = bim2a_advection_diffusion (mesh, epsilon, 1, 1, phi);
Mass    = bim2a_reaction (mesh, 1, 1);
b       = bim2a_rhs (mesh,f,g);
A       = AdvDiff + Mass;
</syntaxhighlight>}}

<b> To Apply Boundary Conditions, partition LHS and RHS</b>

The tags of the sides are assigned by gmsh we let <math> \Gamma_D </math> be composed by sides 1 and 2
and <math> \Gamma_D </math> be the rest of the boundary

{{Code|Boundary conditions for the 2D problem|<syntaxhighlight lang="octave" style="font-size:13px"> 
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); 
</syntaxhighlight>}}


<B> Solve for the displacements</B>

{{Code|Displacement in the 2D problem|<syntaxhighlight lang="octave" style="font-size:13px"> 
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);
</syntaxhighlight>}}

<b> Compute the fluxes through Dirichlet sides</b><br>

{{Code|Fluxes at sides in the 2D problem|<syntaxhighlight lang="octave" style="font-size:13px"> 
jd = [Add Adi Adn] * u([GammaD; Omega; GammaN]) - bd;
</syntaxhighlight>}}


<B> Compute the gradient of the solution </B>

{{Code|Gradient of solution in the 2D problem|<syntaxhighlight lang="octave" style="font-size:13px"> 
[gx, gy] = bim2c_pde_gradient (mesh, u);
</syntaxhighlight>}}

<B> Compute the internal Advection-Diffusion flux</B>

<pre>
[jxglob, jyglob] = bim2c_global_flux (mesh, u, epsilon*ones(nelems, 1), ones(nnodes, 1), ones(nnodes, 1), phi);
</pre>

<B> Export data to VTK format</B>

The resut can be exported to vtk format to visualize with [[http://www.paraview.org|paraview]] 
or [[https://wci.llnl.gov/codes/visit/|visit]]

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

you can also plot your data directly in Octave using <code> pdesurf </code>

<pre>
pdesurf (mesh.p, mesh.t, u)
</pre>

it will look like this

[[File:fiume_sol_pdesurf.png|500px]]

=== 3D Time dependent problem ===

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

The equation being solved is

<math> \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] </math>

<math> ~\varphi = x + y - z </math>

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

The initial condition is

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

{{Code|Define the 3D problem|<syntaxhighlight lang="octave" style="font-size:13px"> 
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
</syntaxhighlight>}}

[http://youtu.be/2E6Z_AcV8CQ This is a video] showing the .3 isosurface of the solution.

== External links ==

* [https://octave.sourceforge.io/bim/index.html BIM package at Octave Forge].



== Scientific papers using BIM ==

* [https://doi.org/10.1016/j.jcp.2004.10.029 de Falco, C., Gatti, E., Lacaita, A. L., & Sacco, R. (2005). Quantum-corrected drift-diffusion models for transport in semiconductor devices. Journal of Computational Physics, 204(2), 533-561.]

* [https://doi.org/10.1016/j.jcp.2008.11.010 de Falco, C., Jerome, J.W. and Sacco, R., 2009. Quantum-corrected drift-diffusion models: Solution fixed point map and finite element approximation. Journal of Computational Physics, 228(5), pp.1770-1789.]

* [http://www.math.ualberta.ca/ijnam/Volume-7-2010/No-3-10/2010-03-04.pdf de Falco, C. and O'riordan, E., 2010. INTERIOR LAYERS IN A REACTION DIFFUSION EQUATION WITH A DISCONTINUOUS DIFFUSION COEFFICIENT. International Journal of Numerical Analysis & Modeling, 7(3).]

* [https://doi.org/10.1016/j.cma.2010.01.018 de Falco, C., Sacco, R. and Verri, M., 2010. Analytical and numerical study of photocurrent transients in organic polymer solar cells. Computer Methods in Applied Mechanics and Engineering, 199(25), pp.1722-1732.]

* [https://doi.org/10.1016/j.cma.2012.06.018 de Falco, C., Porro, M., Sacco, R. and Verri, M., 2012. Multiscale modeling and simulation of organic solar cells. Computer Methods in Applied Mechanics and Engineering, 245, pp.102-116.]

* [http://link.springer.com/chapter/10.1007/978-3-540-71980-9_5 de Falco, C., Denk, G. and Schultz, R., 2007. A demonstrator platform for coupled multiscale simulation. In Scientific Computing in Electrical Engineering (pp. 63-71). Springer Berlin Heidelberg.]

* [https://doi.org/10.1016/j.orgel.2014.12.001 Maddalena, F., de Falco, C., Caironi, M. and Natali, D., 2015. Assessing the width of Gaussian density of states in organic semiconductors. Organic Electronics, 17, pp.304-318.]

* [http://link.springer.com/chapter/10.1007/978-3-642-12294-1_35 Alì, G., Bartel, A., Culpo, M. and de Falco, C., 2010. Analysis of a PDE thermal element model for electrothermal circuit simulation. In Scientific Computing in Electrical Engineering SCEE 2008 (pp. 273-280). Springer Berlin Heidelberg.]

* [http://onlinelibrary.wiley.com/doi/10.1002/pamm.200810065/full Culpo, M. and De Falco, C., 2008. Dynamical iteration schemes for coupled simulation in nanoelectronics. PAMM, 8(1), pp.10065-10068.]

* [https://doi.org/10.1108/COMPEL-12-2012-0365 Porro, M., de Falco, C., Verri, M., Lanzani, G. and Sacco, R., 2014. Multiscale simulation of organic heterojunction light harvesting devices. COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 33(4), pp.1107-1122.]

* [http://dx.doi.org/10.1063/1.4709483 Ciucci, F., de Falco, C., Guzman, M.I., Lee, S. and Honda, T., 2012. Chemisorption on semiconductors: The role of quantum corrections on the space charge regions in multiple dimensions. Applied Physics Letters, 100(18), p.183106.]

* [https://link.springer.com/chapter/10.1007%2F978-3-642-12294-1_36 Culpo, M., de Falco, C., Denk, G. and Voigtmann, S., 2010. Automatic thermal network extraction and multiscale electro-thermal simulation. In Scientific Computing in Electrical Engineering SCEE 2008 (pp. 281-288). Springer Berlin Heidelberg.]

* [https://link.springer.com/chapter/10.1007/978-3-642-12110-4_33 Culpo, M., de Falco, C. and O’Riordan, E., 2010. Patches of finite elements for singularly-perturbed diffusion reaction equations with discontinuous coefficients. In Progress in Industrial Mathematics at ECMI 2008 (pp. 235-240). Springer Berlin Heidelberg.]

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