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== 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. | This is a short example on how to use <tt>bim</tt> to solve a 2D Diffusion Advection Reaction problem. | ||
The coplete code for this example can is on [[Agora]] [[http://agora.octave.org/snippet/1bqV/|here]]. | |||
We want to solve the equation | 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 | <math> -\mathrm{div}\ ( \varepsilon\ \nabla u(x, y) - \nabla \varphi(x,y)\ u(x, y) ) ) + u(x, y) = 1 \qquad in \Omega</math> | ||
<math> \varphi(x, y)\ =\ x + y </math> | <math> \varphi(x, y)\ =\ x + y </math> | ||
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on the mesh structure | on the mesh structure | ||
<pre> | |||
[mesh] = msh2m_gmsh ("fiume","scale",1,"clscale",.1); | [mesh] = msh2m_gmsh ("fiume","scale",1,"clscale",.1); | ||
[mesh] = bim2c_mesh_properties (mesh); | [mesh] = bim2c_mesh_properties (mesh); | ||
</ | </pre> | ||
to see the mesh you can use functions from the [[fpl_package|fpl package]] | to see the mesh you can use functions from the [[fpl_package|fpl package]] | ||
<pre> | |||
pdemesh (mesh.p, mesh.e, mesh.t) | pdemesh (mesh.p, mesh.e, mesh.t) | ||
view (2) | view (2) | ||
</ | </pre> | ||
[[File:fiume_msh.png]] | [[File:fiume_msh.png]] | ||
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Get the node coordinates from the mesh structure | Get the node coordinates from the mesh structure | ||
<pre> | |||
xu = mesh.p(1,:).'; | xu = mesh.p(1,:).'; | ||
yu = mesh.p(2,:).'; | yu = mesh.p(2,:).'; | ||
</ | </pre> | ||
Get the number of elements and nodes in the mesh | Get the number of elements and nodes in the mesh | ||
<pre> | |||
nelems = columns (mesh.t); | nelems = columns (mesh.t); | ||
nnodes = columns (mesh.p); | nnodes = columns (mesh.p); | ||
</ | </pre> | ||
<pre> | |||
epsilon = .1; | epsilon = .1; | ||
phi = xu + yu; | phi = xu + yu; | ||
</ | </pre> | ||
<b> Construct the discretized operators</b> | <b> Construct the discretized operators</b> | ||
<pre> | |||
AdvDiff = bim2a_advection_diffusion (mesh, epsilon, 1, 1, phi); | AdvDiff = bim2a_advection_diffusion (mesh, epsilon, 1, 1, phi); | ||
Mass = bim2a_reaction (mesh, 1, 1); | Mass = bim2a_reaction (mesh, 1, 1); | ||
b = bim2a_rhs (mesh,f,g); | b = bim2a_rhs (mesh,f,g); | ||
A = AdvDiff + Mass; | A = AdvDiff + Mass; | ||
</ | </pre> | ||
<b> To Apply Boundary Conditions, partition LHS and RHS</b> | <b> To Apply Boundary Conditions, partition LHS and RHS</b> | ||
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and <math> \Gamma_D </math> be the rest of the boundary | and <math> \Gamma_D </math> be the rest of the boundary | ||
<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 | ||
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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> | |||
Add = A(GammaD, GammaD); | Add = A(GammaD, GammaD); | ||
Adn = A(GammaD, GammaN); ## shoud be all zeros hopefully!! | Adn = A(GammaD, GammaN); ## shoud be all zeros hopefully!! | ||
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bn = b(GammaN); | bn = b(GammaN); | ||
bi = b(Omega); | bi = b(Omega); | ||
</ | </pre> | ||
<B> Solve for the displacements</B> | |||
< | <pre> | ||
temp = [Ann Ani ; Ain Aii ] \ [ jn+bn-And*ud(GammaD) ; bi-Aid*ud(GammaD)]; | temp = [Ann Ani ; Ain Aii ] \ [ jn+bn-And*ud(GammaD) ; bi-Aid*ud(GammaD)]; | ||
u = ud; | u = ud; | ||
u(GammaN) = temp(1:numel (GammaN)); | u(GammaN) = temp(1:numel (GammaN)); | ||
u(Omega) = temp(length(GammaN)+1:end); | u(Omega) = temp(length(GammaN)+1:end); | ||
</ | </pre> | ||
<b> Compute the fluxes through Dirichlet sides</b><br> | <b> Compute the fluxes through Dirichlet sides</b><br> | ||
<pre> | |||
jd = [Add Adi Adn] * u([GammaD; Omega; GammaN]) - bd; | jd = [Add Adi Adn] * u([GammaD; Omega; GammaN]) - bd; | ||
</ | </pre> | ||
<B> Compute the gradient of the solution </B> | <B> Compute the gradient of the solution </B> | ||
<pre> | |||
[gx, gy] = bim2c_pde_gradient (mesh, u); | [gx, gy] = bim2c_pde_gradient (mesh, u); | ||
</ | </pre> | ||
<B> Compute the internal Advection-Diffusion flux</B> | <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); | [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> | <B> Export data to VTK format</B> | ||
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or [[https://wci.llnl.gov/codes/visit/|visit]] | or [[https://wci.llnl.gov/codes/visit/|visit]] | ||
<pre> | |||
fpl_vtk_write_field ("vtkdata", mesh, {u, "Solution"}, {[gx; gy]', "Gradient"}, 1); | 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> | you can also plot your data directly in Octave using <code> pdesurf </code> | ||
<pre> | |||
pdesurf (mesh.p, mesh.t, u) | pdesurf (mesh.p, mesh.t, u) | ||
</ | </pre> | ||
it will look like this | it will look like this | ||
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[[File:fiume_sol_pdesurf.png|500px]] | [[File:fiume_sol_pdesurf.png|500px]] | ||
[[Category: | [[Category:OctaveForge]] | ||
[[Category:Packages]] |