bim/ 000755 000765 000000 00000000000 12420212271 012054 5 ustar 00carlo wheel 000000 000000 bim/COPYING 000644 000765 000000 00000043077 10751627243 013140 0 ustar 00carlo wheel 000000 000000 GNU GENERAL PUBLIC LICENSE
Version 2, June 1991
Copyright (C) 1989, 1991 Free Software Foundation, Inc.
Everyone is permitted to copy and distribute verbatim copies
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Preamble
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0. This License applies to any program or other work which contains
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How to Apply These Terms to Your New Programs
If you develop a new program, and you want it to be of the greatest
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free software which everyone can redistribute and change under these terms.
To do so, attach the following notices to the program. It is safest
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convey the exclusion of warranty; and each file should have at least
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Copyright (C)
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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If the program is interactive, make it output a short notice like this
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Gnomovision version 69, Copyright (C) year name of author
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This is free software, and you are welcome to redistribute it
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You should also get your employer (if you work as a programmer) or your
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Yoyodyne, Inc., hereby disclaims all copyright interest in the program
`Gnomovision' (which makes passes at compilers) written by James Hacker.
, 1 April 1989
Ty Coon, President of Vice
This General Public License does not permit incorporating your program into
proprietary programs. If your program is a subroutine library, you may
consider it more useful to permit linking proprietary applications with the
library. If this is what you want to do, use the GNU Library General
Public License instead of this License.
bim/DESCRIPTION 000644 000765 000000 00000000566 12420212220 013563 0 ustar 00carlo wheel 000000 000000 Name: bim
Version: 1.1.5
Date: 2014-10-17
Author: Carlo de Falco, Culpo Massimiliano, Matteo Porro, Emanuela Abbate
Maintainer: Carlo de Falco
Title: PDE Solver using a Finite Element/Finite Volume approach
Description: Package for solving Diffusion Advection Reaction (DAR) Partial Differential Equations
Depends: octave (>= 3.8.0), fpl, msh
Autoload: no
License: GPLv2+
bim/doc/ 000755 000765 000000 00000000000 12420212271 012621 5 ustar 00carlo wheel 000000 000000 bim/INDEX 000644 000765 000000 00000002266 12316045351 012663 0 ustar 00carlo wheel 000000 000000 BIM >> BIM - Diffusion Advection Reaction PDE Solver
Matrix assembly
bim1a_advection_diffusion
bim1a_advection_upwind
bim1a_axisymmetric_advection_diffusion
bim1a_axisymmetric_advection_upwind
bim2a_advection_diffusion
bim2a_advection_upwind
bim2a_axisymmetric_advection_diffusion
bim2a_axisymmetric_advection_upwind
bim3a_advection_diffusion
bim3a_osc_advection_diffusion
bim1a_laplacian
bim1a_axisymmetric_laplacian
bim2a_laplacian
bim2a_axisymmetric_laplacian
bim3a_laplacian
bim3a_osc_laplacian
bim1a_reaction
bim1a_axisymmetric_reaction
bim2a_reaction
bim2a_axisymmetric_reaction
bim3a_reaction
bim1a_rhs
bim1a_axisymmetric_rhs
bim2a_rhs
bim2a_axisymmetric_rhs
bim3a_rhs
bim2a_boundary_mass
bim2a_axisymmetric_boundary_mass
bim3a_boundary_mass
Pre-processing and Post-processing computations
bim2c_mesh_properties
bim3c_mesh_properties
bim2c_unknowns_on_side
bim3c_unknowns_on_faces
bim2c_pde_gradient
bim3c_pde_gradient
bim2c_global_flux
bim3c_global_flux
bim1c_elem_to_nodes
bim2c_tri_to_nodes
bim3c_tri_to_nodes
bim2c_intrp
bim3c_intrp
bim1c_norm
bim2c_norm
bim3c_norm
Utilities
bimu_bernoulli
bimu_logm
bim/inst/ 000755 000765 000000 00000000000 12420212272 013032 5 ustar 00carlo wheel 000000 000000 bim/NEWS 000644 000765 000000 00000003311 12420212154 012551 0 ustar 00carlo wheel 000000 000000 Summary of important user-visible changes for bim 1.1.5:
-------------------------------------------------------------------
** Improvement of the functions for stiffness matrix assembly in 2D
axisymmetric configuration.
Summary of important user-visible changes for bim 1.1.4:
-------------------------------------------------------------------
** Added new functions for cylindrical coordinates in axisymmetric
configuration.
** Fixed some bugs in 1d upwind discretization.
Summary of important user-visible changes for bim 1.1.3:
-------------------------------------------------------------------
** Added new projection functions.
** A function for the generation of 3D boundary matrices has been added that
can be used in the implementation of Robin boundary condition and interface
conditions.
** Added a set of functions for the computation of inf, L2 and H1 norms of
piecewise constant and elementwise constant functions.
Summary of important user-visible changes for bim 1.1.2:
-------------------------------------------------------------------
** Fixed a bug in 1d upwind discretization.
** Added function to compute gradient of a 3d scalar field.
Summary of important user-visible changes for bim 1.1.1:
-------------------------------------------------------------------
** Added new functions to perform interpolation at arbitrary nodes.
Summary of important user-visible changes for bim 1.1.0:
-------------------------------------------------------------------
** Added new functions implementing the Orthogonal Subdomain Collocation
method in 3D.
** More functions for 3d problems were added.
** Demostrations of how to use the package are now in the wiki.
bim/inst/bim1a_advection_diffusion.m 000644 000765 000000 00000007717 12316037504 020326 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {[@var{A}]} = @
## bim1a_advection_diffusion(@var{mesh},@var{alpha},@var{gamma},@var{eta},@var{beta})
##
## Build the Scharfetter-Gummel stabilized stiffness matrix for a
## diffusion-advection problem.
##
## The equation taken into account is:
##
## - div (@var{alpha} * @var{gamma} (@var{eta} grad (u) - @var{beta} u)) = f
##
## where @var{alpha} is an element-wise constant scalar function,
## @var{eta} and @var{gamma} are piecewise linear conforming scalar
## functions, @var{beta} is an element-wise constant vector function.
##
## Instead of passing the vector field @var{beta} directly one can pass
## a piecewise linear conforming scalar function @var{phi} as the last
## input. In such case @var{beta} = grad @var{phi} is assumed.
##
## If @var{phi} is a single scalar value @var{beta} is assumed to be 0
## in the whole domain.
##
## @seealso{bim1a_rhs, bim1a_reaction, bim1a_laplacian, bim2a_advection_diffusion}
## @end deftypefn
function A = bim1a_advection_diffusion (x,alpha,gamma,eta,beta)
## Check input
if (nargin != 5)
error ("bim1a_advection_diffusion: wrong number of input parameters.");
elseif (! isvector (x))
error ("bim1a_advection_diffusion: first argument is not a valid vector.");
endif
nnodes = numel (x);
nelem = nnodes - 1;
## Turn scalar input to a vector of appropriate size
if (isscalar (alpha))
alpha = alpha * ones (nelem, 1);
endif
if (isscalar (gamma))
gamma = gamma * ones (nnodes, 1);
endif
if (isscalar (eta))
eta = eta * ones (nnodes, 1);
endif
if (! (isvector (alpha) && isvector (gamma) && isvector (eta)))
error ("bim1a_advection_diffusion: coefficients are not valid vectors.");
elseif (numel (alpha) != nelem)
error ("bim1a_advection_diffusion: length of alpha is not equal to the number of elements.");
elseif (numel (gamma) != nnodes)
error ("bim1a_advection_diffusion: length of gamma is not equal to the number of nodes.");
elseif (numel (eta) != nnodes)
error ("bim1a_advection_diffusion: length of eta is not equal to the number of nodes.");
endif
areak = reshape (diff (x), [], 1);
if (numel (beta) == 1)
vk = 0;
elseif (numel (beta) == nelem)
vk = beta .* areak;
elseif (numel (beta) == nnodes)
vk = diff (beta);
else
error ("bim1a_advection_diffusion: coefficient beta has wrong dimensions.");
endif
gammaetak = bimu_logm ((gamma .* eta)(1:end-1), (gamma .* eta)(2:end));
veta = diff (eta);
etak = bimu_logm (eta(1:end-1), eta(2:end));
ck = alpha .* gammaetak .* etak ./ areak;
[bpk, bmk] = bimu_bernoulli ((vk - veta) ./ etak);
dm1 = [-(ck.*bmk); NaN];
dp1 = [NaN; -(ck.*bpk)];
d0 = [(ck(1).*bmk(1)); ((ck.*bmk)(2:end) + (ck.*bpk)(1:end-1)); (ck(end).*bpk(end))];
A = spdiags([dm1, d0, dp1],-1:1,nnodes,nnodes);
endfunction
%!test
%! x = linspace(0,1,101);
%! A = bim1a_advection_diffusion(x,1,1,1,0);
%! alpha = ones(100,1);
%! gamma = ones(101,1);
%! eta = gamma;
%! B = bim1a_advection_diffusion(x,alpha,gamma,eta,0);
%! assert(A,B)
bim/inst/bim1a_advection_upwind.m 000644 000765 000000 00000006075 12316045351 017641 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2010-2014 Carlo de Falco
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## author: Matteo Porro
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {[@var{A}]} = bim1a_advection_upwind (@var{mesh}, @var{beta})
##
## Build the UW stabilized stiffness matrix for an advection problem.
##
## The equation taken into account is:
##
## (@var{beta} u)' = f
##
## where @var{beta} is an element-wise constant.
##
## Instead of passing the vector field @var{beta} directly one can pass
## a piecewise linear conforming scalar function @var{phi} as the last
## input. In such case @var{beta} = grad @var{phi} is assumed.
##
## If @var{phi} is a single scalar value @var{beta} is assumed to be 0
## in the whole domain.
##
## @seealso{bim1a_rhs, bim1a_reaction, bim1a_laplacian, bim2a_advection_diffusion}
## @end deftypefn
function A = bim1a_advection_upwind (x, beta)
## Check input
if nargin != 2
error("bim1a_advection_upwind: wrong number of input parameters.");
endif
nnodes = length(x);
nelem = nnodes-1;
if (length(beta) == 1)
vk = zeros(nelem,1);
elseif (length(beta) == nelem)
vk = beta;
elseif (length(beta) == nnodes)
vk = diff(beta);
else
error("bim1a_advection_upwind: coefficient beta has wrong dimensions.");
endif
bmk = (vk+abs(vk))/2;
bpk = -(vk-abs(vk))/2;
dm1 = [-bmk; NaN];
dp1 = [NaN; -bpk];
d0 = [bmk(1); bmk(2:end) + bpk(1:end-1); bpk(end)];
A = spdiags([dm1, d0, dp1],-1:1,nnodes,nnodes);
endfunction
%!test
%! n = 200;
%! mesh = linspace(0,1,n+1)';
%! uex = @(r) - r.^2 + 1;
%! Nnodes = numel(mesh);
%! Nelements = Nnodes-1;
%! D = 1; v = 1; sigma = 0;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! beta = 1/D*v*ones(Nelements,1);
%! delta = ones(Nelements,1);
%! zeta = sigma*ones(Nnodes,1);
%! f = @(r) 2*D - 2*v.*r + sigma*uex(r);
%! rhs = bim1a_rhs(mesh, ones(Nelements,1), f(mesh));
%! S = bim1a_laplacian(mesh,alpha,gamma);
%! A = bim1a_advection_upwind(mesh, beta);
%! R = bim1a_reaction(mesh, delta, zeta);
%! S += (A+R);
%! u = zeros(Nnodes,1); u([1 end]) = uex(mesh([1 end]));
%! u(2:end-1) = S(2:end-1,2:end-1)\(rhs(2:end-1) - S(2:end-1,[1 end])*u([1 end]));
%! assert(u,uex(mesh),1e-3)
bim/inst/bim1a_axisymmetric_advection_diffusion.m 000644 000765 000000 00000015156 12316045351 023117 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006-2014 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## author: Matteo Porro
## author: Emanuela Abbate
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {[@var{A}]} = @
## bim1a_axisymmetric_advection_diffusion(@var{mesh},@var{alpha},@var{gamma},@var{eta},@var{beta})
##
## Build the Scharfetter-Gummel stabilized stiffness matrix for a
## diffusion-advection problem in cylindrical coordinates with axisymmetric
## configuration. Rotational symmetry is assumed with respect to be the vertical
## axis r=0. Only grids that DO NOT contain r=0 are admissible.
##
##@example
##@group
## | |-------| OK |-------| | OK |--|-----| NO!
## r=0 r=0 r=0
#@end group
#@end example
##
## The equation taken into account is:
##
## - 1/r * d/dr (@var{alpha} * @var{gamma} (@var{eta} du/dr - @var{beta} u)) = f
##
## where @var{alpha} is an element-wise constant scalar function,
## @var{eta} and @var{gamma} are piecewise linear conforming scalar
## functions, @var{beta} is an element-wise constant vector function.
##
## Instead of passing the vector field @var{beta} directly one can pass
## a piecewise linear conforming scalar function @var{phi} as the last
## input. In such case @var{beta} = grad @var{phi} is assumed.
##
## If @var{phi} is a single scalar value @var{beta} is assumed to be 0
## in the whole domain.
##
## @seealso{bim1a_axisymmetric_rhs, bim1a_axisymmetric_reaction,
## bim1a_axisymmetric_laplacian, bim2a_axisymmetric_advection_diffusion}
## @end deftypefn
function A = bim1a_axisymmetric_advection_diffusion (x,alpha,gamma,eta,beta)
## Check input
if nargin != 5
error("bim1a_axisymmetric_advection_diffusion: wrong number of input parameters.");
elseif !isvector(x)
error("bim1a_axisymmetric_advection_diffusion: first argument is not a valid vector.");
endif
nnodes = length(x);
nelem = nnodes-1;
## Turn scalar input to a vector of appropriate size
if isscalar(alpha)
alpha = alpha*ones(nelem,1);
endif
if isscalar(gamma)
gamma = gamma*ones(nnodes,1);
endif
if isscalar(eta)
eta = eta*ones(nnodes,1);
endif
if !( isvector(alpha) && isvector(gamma) && isvector(eta) )
error("bim1a_axisymmetric_advection_diffusion: coefficients are not valid vectors.");
elseif (length(alpha) != nelem)
error("bim1a_axisymmetric_advection_diffusion: length of alpha is not equal to the number of elements.");
elseif (length(gamma) != nnodes)
error("bim1a_axisymmetric_advection_diffusion: length of gamma is not equal to the number of nodes.");
elseif (length(eta) != nnodes)
error("bim1a_axisymmetric_advection_diffusion: length of eta is not equal to the number of nodes.");
endif
areak = reshape(diff(x),[],1);
cm = reshape((x(1:end-1)+x(2:end))/2,[],1);
if (length(beta) == 1)
vk = 0;
elseif (length(beta) == nelem)
vk = beta .* areak;
elseif (length(beta) == nnodes)
vk = diff(beta);
else
error("bim1a_axisymmetric_advection_diffusion: coefficient beta has wrong dimensions.");
endif
gammaetak = bimu_logm ( (gamma.*eta)(1:end-1), (gamma.*eta)(2:end));
veta = diff(eta);
etak = bimu_logm ( eta(1:end-1), eta(2:end));
ck = alpha .* gammaetak .* etak ./ areak .* abs(cm);
[bpk, bmk] = bimu_bernoulli( (vk - veta)./etak);
dm1 = [-(ck.*bmk); NaN];
dp1 = [NaN; -(ck.*bpk)];
d0 = [(ck(1).*bmk(1)); ((ck.*bmk)(2:end) + (ck.*bpk)(1:end-1)); (ck(end).*bpk(end))];
A = spdiags([dm1, d0, dp1],-1:1,nnodes,nnodes);
endfunction
%!test
%! n = 3;
%! mesh = linspace(1,2,n+1)';
%! uex = @(r) exp(r);
%! duexdr = @(r) uex(r);
%! d2uexdr2 = @(r) uex(r);
%! Nnodes = numel(mesh);
%! Nelements = Nnodes-1;
%! D = 1; v = 1;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! beta = 1/D*v*ones(Nelements,1);
%! f = @(r) -D./r.*duexdr(r) - D.*d2uexdr2(r) ...
%! + v./r .* uex(r) + v * duexdr(r);
%! rhs = bim1a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh));
%! S = bim1a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1); u([1,end]) = uex(mesh([1 end]));
%! u(2:end-1) = S(2:end-1,2:end-1)\(rhs(2:end-1) - S(2:end-1,[1 end])*u([1 end]));
%! assert(u,uex(mesh),1e-7)
%!test
%! n = 100;
%! mesh = linspace(0,1,n+1)';
%! cm = (mesh(1:end-1) + mesh(2:end))/2;
%! uex = @(r) - r.^2 + 1;
%! Nnodes = numel(mesh);
%! Nelements = Nnodes-1;
%! D = 1; v = 0;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! beta = v;
%! f = @(r) 4*D;
%! rhs = bim1a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh));
%! S = bim1a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1); u(end) = uex(mesh(end));
%! u(1:end-1) = S(1:end-1,1:end-1)\(rhs(1:end-1) - S(1:end-1,end)*u(end));
%! assert(u,uex(mesh),1e-3)
%!test
%! n = 100;
%! mesh = linspace(0,1,n+1)';
%! cm = (mesh(1:end-1) + mesh(2:end))/2;
%! uex = @(r) - r.^2 + 1;
%! Nnodes = numel(mesh);
%! Nelements = Nnodes-1;
%! D = 1; v = cm;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! beta = 1/D*v;
%! f = @(r) 4*D + 2 - 4*r.^2;
%! rhs = bim1a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh));
%! S = bim1a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1); u(end) = uex(mesh(end));
%! u(1:end-1) = S(1:end-1,1:end-1)\(rhs(1:end-1) - S(1:end-1,end)*u(end));
%! assert(u,uex(mesh),1e-3)
%!test
%! x = linspace(0,1,101);
%! A = bim1a_axisymmetric_advection_diffusion(x,1,1,1,0);
%! alpha = ones(100,1);
%! gamma = ones(101,1);
%! eta = gamma;
%! B = bim1a_axisymmetric_advection_diffusion(x,alpha,gamma,eta,0);
%! assert(A,B)
bim/inst/bim1a_axisymmetric_advection_upwind.m 000644 000765 000000 00000006571 12316045351 022440 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2010-2014 Carlo de Falco
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## author: Matteo porro
## author: Emanuela Abbate
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {[@var{A}]} = bim1a_axisymmetric_advection_upwind (@var{mesh}, @var{beta})
##
## Build the Upwind stabilized stiffness matrix for an advection problem
## in cylindrical coordinates with axisymmetric configuration.
##
## The equation taken into account is:
##
## 1/r * (r * @var{beta} u)' = f
##
## where @var{beta} is an element-wise constant.
##
## Instead of passing the vector field @var{beta} directly one can pass
## a piecewise linear conforming scalar function @var{phi} as the last
## input. In such case @var{beta} = grad @var{phi} is assumed.
##
## If @var{phi} is a single scalar value @var{beta} is assumed to be 0
## in the whole domain.
##
## @seealso{bim1a_axisymmetric_advection_diffusion, bim1a_axisymmetric_rhs,
## bim1a_axisymmetric_reaction, bim1a_axisymmetric_laplacian}
## @end deftypefn
function A = bim1a_axisymmetric_advection_upwind (x, beta)
## Check input
if nargin != 2
error("bim1a_axisymmetric_advection_upwind: wrong number of input parameters.");
endif
nnodes = length(x);
nelem = nnodes-1;
cm = reshape((x(1:end-1)+x(2:end))/2,[],1)
if (length(beta) == 1)
vk = 0;#zeros(nelem,1);
elseif (length(beta) == nelem)
vk = beta;
elseif (length(beta) == nnodes)
vk = diff(beta);
else
error("bim1a_axisymmetric_advection_upwind: coefficient beta has wrong dimensions.");
endif
bmk = (vk+abs(vk))/2 .* abs(cm);
bpk = -(vk-abs(vk))/2 .* abs(cm);
dm1 = [-bmk; NaN];
dp1 = [NaN; -bpk];
d0 = [bmk(1); bmk(2:end) + bpk(1:end-1); bpk(end)];
A = spdiags([dm1, d0, dp1],-1:1,nnodes,nnodes);
endfunction
%!test
%! nn = 20;
%! mesh = linspace(1,2,nn+1)';
%! D = 1; v = 0; sigma = 0;
%! uex = @(r) exp(r);
%! duexdr = @(r) uex(r);
%! d2uexdr2 = @(r) uex(r);
%! f = @(r,z) -D./r.*duexdr(r) - D.*d2uexdr2(r) ...
%! + v./r .* uex(r) + v * duexdr(r) ...
%! + sigma * uex(r);
%! uex_left = uex(mesh(1)); uex_right = uex(mesh(end));
%! Ar = bim1a_axisymmetric_laplacian (mesh, D, 1);
%! Adv = bim1a_axisymmetric_advection_upwind (mesh, v*ones(nn,1));
%! R = bim1a_axisymmetric_reaction (mesh, sigma, 1);
%! M = Ar + Adv + R;
%! M(1,:) *= 0; M(1,1) = 1;
%! M(end,:) *= 0; M(end, end) = 1;
%! rhs = bim1a_axisymmetric_rhs (mesh, 1, f(mesh));
%! rhs(1) = uex_left; rhs(end) = uex_right;
%! uh = M \ rhs;
%! assert(uh, uex(mesh), 1e-3);
bim/inst/bim1a_axisymmetric_laplacian.m 000644 000765 000000 00000004701 12316045351 021013 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006-2014 Carlo de Falco
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## author: Matteo Porro
## author: Emanuela Abbate
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {@var{A}} = bim1a_axisymmetric_laplacian (@var{mesh},@var{epsilon},@var{kappa})
##
## Build the standard finite element stiffness matrix for a diffusion
## problem in cylindrical coordinates with axisymmetric configuration.
## Rotational symmetry is assumed with respect to be the vertical
## axis r=0. Only grids that DO NOT contain r=0 are admissible.
##
##@example
##@group
## | |-------| OK |--|-----| NO!
## r=0 r=0
#@end group
#@end example
##
## The equation taken into account is:
##
## - 1/r * (r * @var{epsilon} * @var{kappa} ( u' ))' = f
##
## where @var{epsilon} is an element-wise constant scalar function,
## while @var{kappa} is a piecewise linear conforming scalar function.
##
## @seealso{bim1a_axisymmetric_rhs, bim1a_axisymmetric_reaction,
## bim1a_axisymmetric_advection_diffusion, bim2a_laplacian, bim3a_laplacian}
## @end deftypefn
function [A] = bim1a_axisymmetric_laplacian(mesh,epsilon,kappa)
## Check input
if nargin != 3
error("bim1a_axisymmetric_laplacian: wrong number of input parameters.");
elseif !isvector(mesh)
error("bim1a_axisymmetric_laplacian: first argument is not a valid vector.");
endif
## Input-type check inside bim1a_axisymmetric_advection_diffusion
nnodes = length(mesh);
nelem = nnodes - 1;
A = bim1a_axisymmetric_advection_diffusion (mesh, epsilon, kappa, ones(nnodes,1), 0);
endfunction
bim/inst/bim1a_axisymmetric_reaction.m 000644 000765 000000 00000010727 12316045351 020700 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006-2014 Carlo de Falco
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## author: Matteo Porro
## author: Emanuela Abbate
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{C}]} = @
## bim1a_axisymmetric_reaction(@var{mesh},@var{delta},@var{zeta})
##
## Build the lumped finite element mass matrix for a diffusion
## problem in cylindrical coordinates with axisymmetric configuration.
##
## The equation taken into account is:
##
## @var{delta} * @var{zeta} * u = f
##
## where @var{delta} is an element-wise constant scalar function, while
## @var{zeta} is a piecewise linear conforming scalar function.
##
## @seealso{bim1a_axisymmetric_rhs, bim1a_axisymmetric_advection_diffusion, bim1a_axisymmetric_laplacian,
## bim2a_reaction, bim3a_reaction}
## @end deftypefn
function [C] = bim1a_axisymmetric_reaction(mesh,delta,zeta)
## Check input
if nargin != 3
error("bim1a_axisymmetric_reaction: wrong number of input parameters.");
elseif !isvector(mesh)
error("bim1a_axisymmetric_reaction: first argument is not a valid vector.");
endif
mesh = reshape(mesh,[],1);
nnodes = length(mesh);
nelems = nnodes-1;
## Turn scalar input to a vector of appropriate size
if isscalar(delta)
delta = delta*ones(nelems,1);
endif
if isscalar(zeta)
zeta = zeta*ones(nnodes,1);
endif
if !( isvector(delta) && isvector(zeta) )
error("bim1a_axisymmetric_reaction: coefficients are not valid vectors.");
elseif length(delta) != nelems
error("bim1a_axisymmetric_reaction: length of delta is not equal to the number of elements.");
elseif length(zeta) != nnodes
error("bim1a_axisymmetric_reaction: length of zeta is not equal to the number of nodes.");
endif
h = (mesh(2:end)-mesh(1:end-1)).*delta;
d0 = zeta.*[h(1)/2; (h(1:end-1)+h(2:end))/2; h(end)/2];
C = spdiags(d0.*abs(mesh), 0, nnodes,nnodes);
endfunction
%!test
%! n = 100;
%! mesh = linspace(0,1,n+1)';
%! cm = (mesh(1:end-1) + mesh(2:end))/2;
%! uex = @(r) - r.^2 + 1;
%! Nnodes = numel(mesh);
%! Nelements = Nnodes-1;
%! D = 1; v = cm; sigma = 1;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! beta = 0;
%! delta = ones(Nelements,1);
%! zeta = sigma*ones(Nnodes,1);
%! f = @(r) 4*D + sigma*uex(r);
%! rhs = bim1a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh));
%! S = bim1a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! R = bim1a_axisymmetric_reaction(mesh, delta, zeta);
%! S += R;
%! u = zeros(Nnodes,1); u(end) = uex(mesh(end));
%! u(1:end-1) = S(1:end-1,1:end-1)\(rhs(1:end-1) - S(1:end-1,end)*u(end));
%! assert(u,uex(mesh),1e-3)
%!test
%! n = 100;
%! mesh = linspace(0,1,n+1)';
%! cm = (mesh(1:end-1) + mesh(2:end))/2;
%! uex = @(r) - r.^2 + 1;
%! Nnodes = numel(mesh);
%! Nelements = Nnodes-1;
%! D = 1; v = cm; sigma = 1;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! beta = 1/D*v;
%! delta = ones(Nelements,1);
%! zeta = sigma*ones(Nnodes,1);
%! f = @(r) 4*D + 2 - 4*r.^2 + sigma*uex(r);
%! rhs = bim1a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh));
%! S = bim1a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! R = bim1a_axisymmetric_reaction(mesh, delta, zeta);
%! S += R;
%! u = zeros(Nnodes,1); u(end) = uex(mesh(end));
%! u(1:end-1) = S(1:end-1,1:end-1)\(rhs(1:end-1) - S(1:end-1,end)*u(end));
%! assert(u,uex(mesh),1e-3)
%!test
%! x = linspace(0,1,101);
%! A = bim1a_axisymmetric_reaction(x,1,1);
%! delta = ones(100,1);
%! zeta = ones(101,1);
%! B = bim1a_axisymmetric_reaction(x,delta,zeta);
%! assert(A,B)
bim/inst/bim1a_axisymmetric_rhs.m 000644 000765 000000 00000005226 12316045351 017666 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006-2014 Carlo de Falco
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## author: Matteo Porro
## author: Emanuela Abbate
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{b}]} = @
## bim1a_rhs(@var{mesh},@var{f}, @var{g})
##
## Build the finite element right-hand side of a diffusion problem
## employing mass-lumping.
##
## The equation taken into account is:
##
## @var{delta} * u = f * g
##
## where @var{f} is an element-wise constant scalar function, while
## @var{g} is a piecewise linear conforming scalar function.
##
## @seealso{bim1a_reaction, bim1a_advection_diffusion, bim1a_laplacian,
## bim2a_reaction, bim3a_reaction}
## @end deftypefn
function b = bim1a_axisymmetric_rhs(mesh,f,g)
## Check input
if nargin != 3
error("bim1a_rad_rhs: wrong number of input parameters.");
elseif !isvector(mesh)
error("bim1a_rad_rhs: first argument is not a valid vector.");
endif
mesh = reshape(mesh,[],1);
nnodes = length(mesh);
nelem = nnodes-1;
## Turn scalar input to a vector of appropriate size
if isscalar(f)
f = f*ones(nelem,1);
endif
if isscalar(g)
g = g*ones(nnodes,1);
endif
if !( isvector(f) && isvector(g) )
error("bim1a_rad_rhs: coefficients are not valid vectors.");
elseif (length(f) != nelem && length(f) != 1)
error("bim1a_rad_rhs: length of f is not equal to the number of elements.");
elseif (length(g) != nnodes && length(g) != 1)
error("bim1a_rad_rhs: length of g is not equal to the number of nodes.");
endif
h = (mesh(2:end)-mesh(1:end-1)).*f;
b = g.*[h(1)/2; (h(1:end-1)+h(2:end))/2; h(end)/2] .* abs(mesh);
endfunction
%!test
%! x = linspace(0,1,101);
%! A = bim1a_axisymmetric_rhs(x,1,1);
%! delta = ones(100,1);
%! zeta = ones(101,1);
%! B = bim1a_axisymmetric_rhs(x,delta,zeta);
%! assert(A,B)
bim/inst/bim1a_laplacian.m 000644 000765 000000 00000003632 12316041364 016217 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {@var{A}} = bim1a_laplacian (@var{mesh},@var{epsilon},@var{kappa})
##
## Build the standard finite element stiffness matrix for a diffusion
## problem.
##
## The equation taken into account is:
##
## - (@var{epsilon} * @var{kappa} ( u' ))' = f
##
## where @var{epsilon} is an element-wise constant scalar function,
## while @var{kappa} is a piecewise linear conforming scalar function.
##
## @seealso{bim1a_rhs, bim1a_reaction, bim1a_advection_diffusion,
## bim2a_laplacian, bim3a_laplacian}
## @end deftypefn
function [A] = bim1a_laplacian(mesh,epsilon,kappa)
## Check input
if nargin != 3
error("bim1a_laplacian: wrong number of input parameters.");
elseif !isvector(mesh)
error("bim1a_laplacian: first argument is not a valid vector.");
endif
## Input-type check inside bim1a_advection_diffusion
nnodes = numel (mesh);
nelem = nnodes - 1;
A = bim1a_advection_diffusion (mesh, epsilon, kappa, ones(nnodes,1), 0);
endfunction
bim/inst/bim1a_reaction.m 000644 000765 000000 00000005137 12316041364 016101 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{C}]} = @
## bim1a_reaction(@var{mesh},@var{delta},@var{zeta})
##
## Build the lumped finite element mass matrix for a diffusion
## problem.
##
## The equation taken into account is:
##
## @var{delta} * @var{zeta} * u = f
##
## where @var{delta} is an element-wise constant scalar function, while
## @var{zeta} is a piecewise linear conforming scalar function.
##
## @seealso{bim1a_rhs, bim1a_advection_diffusion, bim1a_laplacian,
## bim2a_reaction, bim3a_reaction}
## @end deftypefn
function [C] = bim1a_reaction(mesh,delta,zeta)
## Check input
if nargin != 3
error("bim1a_reaction: wrong number of input parameters.");
elseif !isvector(mesh)
error("bim1a_reaction: first argument is not a valid vector.");
endif
mesh = reshape (mesh, [], 1);
nnodes = numel (mesh);
nelems = nnodes - 1;
## Turn scalar input to a vector of appropriate size
if isscalar(delta)
delta = delta*ones(nelems,1);
endif
if isscalar(zeta)
zeta = zeta*ones(nnodes,1);
endif
if !( isvector(delta) && isvector(zeta) )
error("bim1a_reaction: coefficients are not valid vectors.");
elseif numel (delta) != nelems
error("bim1a_reaction: length of delta is not equal to the number of elements.");
elseif numel (zeta) != nnodes
error("bim1a_reaction: length of zeta is not equal to the number of nodes.");
endif
h = (mesh(2:end)-mesh(1:end-1)).*delta;
d0 = zeta.*[h(1)/2; (h(1:end-1)+h(2:end))/2; h(end)/2];
C = spdiags(d0, 0, nnodes,nnodes);
endfunction
%!test
%! x = linspace(0,1,101);
%! A = bim1a_reaction(x,1,1);
%! delta = ones(100,1);
%! zeta = ones(101,1);
%! B = bim1a_reaction(x,delta,zeta);
%! assert(A,B)
bim/inst/bim1a_rhs.m 000644 000765 000000 00000004751 12316041364 015072 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{b}]} = @
## bim1a_rhs(@var{mesh},@var{f}, @var{g})
##
## Build the finite element right-hand side of a diffusion problem
## employing mass-lumping.
##
## The equation taken into account is:
##
## @var{delta} * u = f * g
##
## where @var{f} is an element-wise constant scalar function, while
## @var{g} is a piecewise linear conforming scalar function.
##
## @seealso{bim1a_reaction, bim1a_advection_diffusion, bim1a_laplacian,
## bim2a_reaction, bim3a_reaction}
## @end deftypefn
function b = bim1a_rhs(mesh,f,g)
## Check input
if nargin != 3
error("bim1a_rhs: wrong number of input parameters.");
elseif !isvector(mesh)
error("bim1a_rhs: first argument is not a valid vector.");
endif
mesh = reshape(mesh,[],1);
nnodes = numel (mesh);
nelem = nnodes-1;
## Turn scalar input to a vector of appropriate size
if isscalar(f)
f = f*ones(nelem,1);
endif
if isscalar(g)
g = g*ones(nnodes,1);
endif
if !( isvector(f) && isvector(g) )
error("bim1a_rhs: coefficients are not valid vectors.");
elseif (numel (f) != nelem && numel (f) != 1)
error("bim1a_rhs: length of f is not equal to the number of elements.");
elseif (numel (g) != nnodes && numel (g) != 1)
error("bim1a_rhs: length of g is not equal to the number of nodes.");
endif
h = (mesh(2:end)-mesh(1:end-1)).*f;
b = g.*[h(1)/2; (h(1:end-1)+h(2:end))/2; h(end)/2];
endfunction
%!test
%! x = linspace(0,1,101);
%! A = bim1a_rhs(x,1,1);
%! delta = ones(100,1);
%! zeta = ones(101,1);
%! B = bim1a_rhs(x,delta,zeta);
%! assert(A,B)
bim/inst/bim1c_elem_to_nodes.m 000644 000765 000000 00000005412 12241137657 017117 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2013 Carlo de Falco
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## .
## -*- texinfo -*-
##
## @deftypefn {Function File} {@var{u_nod}} = bim1c_elem_to_nodes (@var{mesh}, @var{u_el})
## @deftypefnx {Function File} {@var{u_nod}} = bim1c_elem_to_nodes (@var{m_el}, @var{u_el})
## @deftypefnx {Function File} {[@var{u_nod}, @var{m_el}]} = bim1c_elem_to_nodes ( ... )
##
## Compute interpolated values at nodes @var{u_nod} given values at element mid-points @var{u_el}.
## If called with more than one output, also return the interpolation matrix @var{m_el} such that
## @code{u_nod = m_el * u_el}.
## If repeatedly performing interpolation on the same mesh the matrix @var{m_el} obtained by a previous call
## to @code{bim1c_elem_to_nodes} may be passed as input to avoid unnecessary computations.
##
## @end deftypefn
## Author: Carlo de Falco
## Author: Matteo Porro
## Created: 2013-11-04
function [u_nod, m_el] = bim1c_elem_to_nodes (m, u_el)
if (nargout > 1 )
if (isvector (m))
nel = numel (m) - 1;
nnod = numel (m);
m_el = spalloc (nnod, nel, 2 * nel);
h = diff (m);
for iel = 1:nel
m_el([iel, iel+1], iel) = h(iel);
endfor
m_el = diag (sum (m_el, 2)) \ m_el;
elseif (ismatrix (m))
m_el = m;
else
error (["bim1c_elem_to_nodes: first input ", ...
"parameter is of incorrect type"]);
endif
u_nod = m_el * u_el;
else
if (isvector (m))
rhs = bim1a_rhs (m, u_el, 1);
mass = bim1a_reaction (m, 1, 1);
u_nod = full (mass \ rhs);
elseif (ismatrix (m))
u_nod = m * u_el;
else
error (["bim1c_elem_to_nodes: first input ", ...
"parameter is of incorrect type"]);
endif
endif
endfunction
%!test
%! n = 10; msh = linspace (0, 1, n+1);
%! nel = n;
%! nnod = n+1;
%! u_el = randn (nel, 1);
%! un1 = bim1c_elem_to_nodes (msh, u_el);
%! [un2, m] = bim1c_elem_to_nodes (msh, u_el);
%! un3 = bim1c_elem_to_nodes (m, u_el);
%! [un4, m] = bim1c_elem_to_nodes (m, u_el);
%! assert (un1, un2, 1e-10)
%! assert (un1, un3, 1e-10)
%! assert (un1, un4, 1e-10)
bim/inst/bim1c_norm.m 000644 000765 000000 00000007226 12316041364 015253 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006-2013 Carlo de Falco
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Matteo Porro
## -*- texinfo -*-
##
## @deftypefn {Function File} {[@var{norm_u}]} = @
## bim1c_norm(@var{mesh},@var{u},@var{norm_type})
##
## Compute the @var{norm_type}-norm of function @var{u} on the domain described
## by the triangular grid @var{mesh}.
##
## The input function @var{u} can be either a piecewise linear conforming scalar
## function or an elementwise constant scalar or vector function.
##
## The string parameter @var{norm_type} can be one among 'L2', 'H1' and 'inf'.
##
## Should the input function be piecewise constant, the H1 norm will not be
## computed and the function will return an error message.
##
## @seealso{bim2c_norm, bim3c_norm}
##
## @end deftypefn
function [norm_u] = bim1c_norm (m, u, norm_type)
## Check input
if (nargin != 3)
error ("bim1c_norm: wrong number of input parameters.");
elseif (! isvector (m))
error ("bim1c_norm: first input is not a valid mesh.");
endif
nnodes = numel (m);
nel = numel (m) - 1;
if (isrow (u))
u = u';
endif
if (isrow (m))
m = m';
endif
if ((numel (u) != nnodes) && (numel (u) != nel))
error ("bim1c_norm: numel(u) != nnodes and numel(u) != nel.");
endif
if (! (strcmp (norm_type, 'L2')
|| strcmp (norm_type, 'inf')
|| strcmp (norm_type, 'H1')))
error ("bim1c_norm: invalid norm type parameter.");
endif
if (strcmp (norm_type,'inf'))
norm_u = max (abs (u));
else
if (numel (u) == nnodes)
M = __mass_matrix__ (m);
if (strcmp (norm_type, 'H1'))
A = bim1a_laplacian (m, 1, 1);
M += A;
endif
norm_u = sqrt(u' * M * u);
else
if (strcmp (norm_type, 'H1'))
error (["bim1c_norm: cannot compute the ", ...
"H1 norm of an elementwise constant function."]);
endif
norm_u = diff(m)' * u.^2;
norm_u = sqrt (norm_u);
endif
endif
endfunction
function M = __mass_matrix__ (m)
nnodes = numel(m);
h = diff(m);
d0 = 1/3*[h(1); h(1:end-1)+h(2:end); h(end)];
d1 = [0; 1/6*h];
dm1 = [1/6*h; 0];
M = spdiags([dm1 d0 d1], -1:1, nnodes, nnodes);
endfunction
%!test
%!shared L, V, m
%! L = rand (1); V = rand (1); m = linspace (0,1,5).^2; m *= L;
%! u = V * ones (size (m))';
%! uinf = bim1c_norm (m, u, 'inf');
%! uL2 = bim1c_norm (m, u, 'L2');
%! uH1 = bim1c_norm (m, u, 'H1');
%! assert ([uinf, uL2, uH1], [V, V*sqrt(L), V*sqrt(L)], 1e-12);
%!test
%! u = V * m';
%! uinf = bim1c_norm (m, u, 'inf');
%! uL2 = bim1c_norm (m, u, 'L2');
%! uH1 = bim1c_norm (m, u, 'H1');
%! assert ([uinf, uL2, uH1],
%! [L*V, V*sqrt(L^3/3), V*sqrt(L^3/3 + L)],
%! 1e-12);
%!test
%! u = V * ones (size (diff (m)))';
%! uinf = bim1c_norm (m, u, 'inf');
%! uL2 = bim1c_norm (m, u, 'L2');
%! assert ([uinf, uL2], [V, V*sqrt(L)], 1e-12);
bim/inst/bim2a_advection_diffusion.m 000644 000765 000000 00000027521 12316041364 020321 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {[@var{A}]} = @
## bim2a_advection_diffusion(@var{mesh},@var{alpha},@var{gamma},@var{eta},@var{beta})
##
## Build the Scharfetter-Gummel stabilized stiffness matrix for a
## diffusion-advection problem.
##
## The equation taken into account is:
##
## - div (@var{alpha} * @var{gamma} (@var{eta} grad (u) - @var{beta} u )) = f
##
## where @var{alpha} is an element-wise constant scalar function,
## @var{eta} and @var{gamma} are piecewise linear conforming scalar
## functions, @var{beta} is an element-wise constant vector function.
##
## Instead of passing the vector field @var{beta} directly one can pass
## a piecewise linear conforming scalar function @var{phi} as the last
## input. In such case @var{beta} = grad @var{phi} is assumed.
##
## If @var{phi} is a single scalar value @var{beta} is assumed to be 0
## in the whole domain.
##
## Example:
## @example
## mesh = msh2m_structured_mesh([0:1/3:1],[0:1/3:1],1,1:4);
## mesh = bim2c_mesh_properties(mesh);
## x = mesh.p(1,:)';
##
## Dnodes = bim2c_unknowns_on_side(mesh,[2,4]);
## Nnodes = columns(mesh.p);
## Nelements = columns(mesh.t);
## Varnodes = setdiff(1:Nnodes,Dnodes);
##
## alpha = ones(Nelements,1);
## eta = .1*ones(Nnodes,1);
## beta = [ones(1,Nelements);zeros(1,Nelements)];
## gamma = ones(Nnodes,1);
## f = bim2a_rhs(mesh,ones(Nnodes,1),ones(Nelements,1));
##
## S = bim2a_advection_diffusion(mesh,alpha,gamma,eta,beta);
## u = zeros(Nnodes,1);
## uex = x - (exp(10*x)-1)/(exp(10)-1);
##
## u(Varnodes) = S(Varnodes,Varnodes)\f(Varnodes);
##
## assert(u,uex,1e-7)
## @end example
##
## @seealso{bim2a_rhs, bim2a_reaction, bim2c_mesh_properties}
## @end deftypefn
function [A] = bim2a_advection_diffusion (mesh, alpha, gamma, eta, beta)
## Check input
if nargin != 5
error("bim2a_advection_diffusion: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim2a_advection_diffusion: first input is not a valid mesh structure.");
endif
nnodes = columns(mesh.p);
nelem = columns(mesh.t);
## Turn scalar input to a vector of appropriate size
if isscalar(alpha)
alpha = alpha*ones(nelem,1);
endif
if isscalar(gamma)
gamma = gamma*ones(nnodes,1);
endif
if isscalar(eta)
eta = eta*ones(nnodes,1);
endif
if !( isvector(alpha) && isvector(gamma) && isvector(eta) )
error("bim2a_advection_diffusion: coefficients are not valid vectors.");
elseif (numel (alpha) != nelem)
error("bim2a_advection_diffusion: length of alpha is not equal to the number of elements.");
elseif (numel (gamma) != nnodes)
error("bim2a_advection_diffusion: length of gamma is not equal to the number of nodes.");
elseif (numel (eta) != nnodes)
error("bim2a_advection_diffusion: length of eta is not equal to the number of nodes.");
endif
alphaareak = reshape (alpha.*mesh.area,1,1,nelem);
shg = mesh.shg(:,:,:);
## Build local Laplacian matrix
Lloc = zeros(3,3,nelem);
for inode = 1:3
for jnode = 1:3
ginode(inode,jnode,:) = mesh.t(inode,:);
gjnode(inode,jnode,:) = mesh.t(jnode,:);
Lloc(inode,jnode,:) = sum( shg(:,inode,:) .* shg(:,jnode,:),1) .* alphaareak;
endfor
endfor
x = mesh.p(1,:);
x = x(mesh.t(1:3,:));
y = mesh.p(2,:);
y = y(mesh.t(1:3,:));
if all(size(beta)==1)
v12 = 0;
v23 = 0;
v31 = 0;
elseif all(size(beta)==[2,nelem])
v12 = beta(1,:) .* (x(2,:)-x(1,:)) + beta(2,:) .* (y(2,:)-y(1,:));
v23 = beta(1,:) .* (x(3,:)-x(2,:)) + beta(2,:) .* (y(3,:)-y(2,:));
v31 = beta(1,:) .* (x(1,:)-x(3,:)) + beta(2,:) .* (y(1,:)-y(3,:));
elseif all(size(beta)==[nnodes,1])
betaloc = beta(mesh.t(1:3,:));
v12 = betaloc(2,:)-betaloc(1,:);
v23 = betaloc(3,:)-betaloc(2,:);
v31 = betaloc(1,:)-betaloc(3,:);
else
error("bim2a_advection_diffusion: coefficient beta has wrong dimensions.");
endif
etaloc = eta(mesh.t(1:3,:));
eta12 = etaloc(2,:) - etaloc(1,:);
eta23 = etaloc(3,:) - etaloc(2,:);
eta31 = etaloc(1,:) - etaloc(3,:);
etalocm1 = bimu_logm(etaloc(2,:),etaloc(3,:));
etalocm2 = bimu_logm(etaloc(3,:),etaloc(1,:));
etalocm3 = bimu_logm(etaloc(1,:),etaloc(2,:));
gammaloc = gamma(mesh.t(1:3,:));
geloc = gammaloc.*etaloc;
gelocm1 = bimu_logm (geloc(2,:), geloc(3,:));
gelocm2 = bimu_logm (geloc(3,:), geloc(1,:));
gelocm3 = bimu_logm (geloc(1,:), geloc(2,:));
[bp12,bm12] = bimu_bernoulli ((v12 - eta12) ./ etalocm3);
[bp23,bm23] = bimu_bernoulli ((v23 - eta23) ./ etalocm1);
[bp31,bm31] = bimu_bernoulli ((v31 - eta31) ./ etalocm2);
bp12 = reshape(gelocm3.*etalocm3.*bp12,1,1,nelem).*Lloc(1,2,:);
bm12 = reshape(gelocm3.*etalocm3.*bm12,1,1,nelem).*Lloc(1,2,:);
bp23 = reshape(gelocm1.*etalocm1.*bp23,1,1,nelem).*Lloc(2,3,:);
bm23 = reshape(gelocm1.*etalocm1.*bm23,1,1,nelem).*Lloc(2,3,:);
bp31 = reshape(gelocm2.*etalocm2.*bp31,1,1,nelem).*Lloc(3,1,:);
bm31 = reshape(gelocm2.*etalocm2.*bm31,1,1,nelem).*Lloc(3,1,:);
Sloc(1,1,:) = (-bm12-bp31)./reshape(etaloc(1,:),1,1,nelem);
Sloc(1,2,:) = bp12./reshape(etaloc(2,:),1,1,nelem);
Sloc(1,3,:) = bm31./reshape(etaloc(3,:),1,1,nelem);
Sloc(2,1,:) = bm12./reshape(etaloc(1,:),1,1,nelem);
Sloc(2,2,:) = (-bp12-bm23)./reshape(etaloc(2,:),1,1,nelem);
Sloc(2,3,:) = bp23./reshape(etaloc(3,:),1,1,nelem);
Sloc(3,1,:) = bp31./reshape(etaloc(1,:),1,1,nelem);
Sloc(3,2,:) = bm23./reshape(etaloc(2,:),1,1,nelem);
Sloc(3,3,:) = (-bm31-bp23)./reshape(etaloc(3,:),1,1,nelem);
A = sparse(ginode(:),gjnode(:),Sloc(:));
endfunction
%!test
%! [mesh] = msh2m_structured_mesh([0:1/3:1],[0:1/3:1],1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! x = mesh.p(1,:)';
%! Dnodes = bim2c_unknowns_on_side(mesh,[2,4]);
%! Nnodes = columns(mesh.p);
%! Nelements = columns(mesh.t);
%! Varnodes = setdiff(1:Nnodes,Dnodes);
%! alpha = ones(Nelements,1);
%! eta = .1*ones(Nnodes,1);
%! beta = [ones(1,Nelements);zeros(1,Nelements)];
%! gamma = ones(Nnodes,1);
%! f = bim2a_rhs(mesh,ones(Nelements,1),ones(Nnodes,1));
%! S = bim2a_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1);
%! u(Varnodes) = S(Varnodes,Varnodes)\f(Varnodes);
%! uex = x - (exp(10*x)-1)/(exp(10)-1);
%! assert(u,uex,1e-7)
%!test
%! [mesh] = msh2m_structured_mesh([0:1/3:1],[0:1/3:1],1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! x = mesh.p(1,:)';
%! Dnodes = bim2c_unknowns_on_side(mesh,[2,4]);
%! Nnodes = columns(mesh.p); Nelements = columns(mesh.t);
%! Varnodes = setdiff(1:Nnodes,Dnodes);
%! alpha = ones(Nelements,1);
%! eta = .1*ones(Nnodes,1);
%! beta = x;
%! gamma = ones(Nnodes,1);
%! f = bim2a_rhs(mesh,ones(Nelements,1),ones(Nnodes,1));
%! S = bim2a_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1);
%! u(Varnodes) = S(Varnodes,Varnodes)\f(Varnodes);
%! uex = x - (exp(10*x)-1)/(exp(10)-1);
%! assert(u,uex,1e-7)
%!test
%! [mesh] = msh2m_structured_mesh([0:1/3:1],[0:1/3:1],1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! x = mesh.p(1,:)';
%! Dnodes = bim2c_unknowns_on_side(mesh,[2,4]);
%! Nnodes = columns(mesh.p); Nelements = columns(mesh.t);
%! Varnodes = setdiff(1:Nnodes,Dnodes);
%! alpha = 10*ones(Nelements,1);
%! eta = .01*ones(Nnodes,1);
%! beta = x/10;
%! gamma = ones(Nnodes,1);
%! f = bim2a_rhs(mesh,ones(Nelements,1),ones(Nnodes,1));
%! S = bim2a_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1);
%! u(Varnodes) = S(Varnodes,Varnodes)\f(Varnodes);
%! uex = x - (exp(10*x)-1)/(exp(10)-1);
%! assert(u,uex,1e-7)
%!test
%! [mesh] = msh2m_structured_mesh([0:1/3:1],[0:1/3:1],1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! x = mesh.p(1,:)';
%! Dnodes = bim2c_unknowns_on_side(mesh,[2,4]);
%! Nnodes = columns(mesh.p); Nelements = columns(mesh.t);
%! Varnodes = setdiff(1:Nnodes,Dnodes);
%! alpha = 10*ones(Nelements,1); eta = .001*ones(Nnodes,1);
%! beta = x/100;
%! gamma = 10*ones(Nnodes,1);
%! f = bim2a_rhs(mesh,ones(Nelements,1),ones(Nnodes,1));
%! S = bim2a_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1);
%! u(Varnodes) = S(Varnodes,Varnodes)\f(Varnodes);
%! uex = x - (exp(10*x)-1)/(exp(10)-1);
%! assert(u,uex,1e-7)
%!test
%! [mesh] = msh2m_structured_mesh([0:1/1e3:1],[0:1/2:1],1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! x = mesh.p(1,:)';
%! Dnodes = bim2c_unknowns_on_side(mesh,[2,4]);
%! Nnodes = columns(mesh.p); Nelements = columns(mesh.t);
%! Varnodes = setdiff(1:Nnodes,Dnodes);
%! alpha = 3*ones(Nelements,1); eta = x+1;
%! beta = [ones(1,Nelements);zeros(1,Nelements)];
%! gamma = 2*x;
%! ff = 2*(6*x.^2+6*x) - (6*x+6).*(1-2*x)+6*(x-x.^2);
%! f = bim2a_rhs(mesh,ones(Nelements,1),ff);
%! S = bim2a_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1);
%! u(Varnodes) = S(Varnodes,Varnodes)\f(Varnodes);
%! uex = x - x.^2;
%! assert(u,uex,5e-3)
%!test
%! [mesh] = msh2m_structured_mesh([0:1/1e3:1],[0:1/2:1],1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! x = mesh.p(1,:)';
%! Dnodes = bim2c_unknowns_on_side(mesh,[2,4]);
%! Nnodes = columns(mesh.p); Nelements = columns(mesh.t);
%! Varnodes = setdiff(1:Nnodes,Dnodes);
%! alpha = ones(Nelements,1); eta = ones(Nnodes,1);
%! beta = 0;
%! gamma = x+1;
%! ff = 4*x+1;
%! f = bim2a_rhs(mesh,ones(Nelements,1),ff);
%! S = bim2a_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1);
%! u(Varnodes) = S(Varnodes,Varnodes)\f(Varnodes);
%! uex = x - x.^2;
%! assert(u,uex,1e-7)
%!test
%! [mesh] = msh2m_structured_mesh([0:.1:1],[0:.1:1],1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! x = mesh.p(1,:)';y = mesh.p(2,:)';
%! Dnodes = bim2c_unknowns_on_side(mesh,[1:4]);
%! Nnodes = columns(mesh.p); Nelements = columns(mesh.t);
%! Varnodes = setdiff(1:Nnodes,Dnodes);
%! alpha = ones(Nelements,1); diff = 1e-2; eta=diff*ones(Nnodes,1);
%! beta =[ones(1,Nelements);ones(1,Nelements)];
%! gamma = x*0+1;
%! ux = y.*(1-exp((y-1)/diff)) .* (1-exp((x-1)/diff)-x.*exp((x-1)/diff)/diff);
%! uy = x.*(1-exp((x-1)/diff)) .* (1-exp((y-1)/diff)-y.*exp((y-1)/diff)/diff);
%! uxx = y.*(1-exp((y-1)/diff)) .* (-2*exp((x-1)/diff)/diff-x.*exp((x-1)/diff)/(diff^2));
%! uyy = x.*(1-exp((x-1)/diff)) .* (-2*exp((y-1)/diff)/diff-y.*exp((y-1)/diff)/(diff^2));
%! ff = -diff*(uxx+uyy)+ux+uy;
%! f = bim2a_rhs(mesh,ones(Nelements,1),ff);
%! S = bim2a_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1);
%! u(Varnodes) = S(Varnodes,Varnodes)\f(Varnodes);
%! uex = x.*y.*(1-exp((x-1)/diff)).*(1-exp((y-1)/diff));
%! assert(u,uex,1e-7)
%!test
%! [mesh] = msh2m_structured_mesh([0:.1:1],[0:.1:1],1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! x = mesh.p(1,:)'; y = mesh.p(2,:)';
%! Dnodes = bim2c_unknowns_on_side(mesh,[1:4]);
%! Nnodes = columns(mesh.p); Nelements = columns(mesh.t);
%! alpha = ones(Nelements,1); eta=ones(Nnodes,1);
%! beta = 0;
%! gamma = ones(Nnodes,1);
%! A = bim2a_advection_diffusion(mesh,1,1,1,0);
%! B = bim2a_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! assert(A,B)
bim/inst/bim2a_advection_upwind.m 000644 000765 000000 00000007532 12041000520 017621 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {[@var{A}]} = @
## bim2a_advection_upwind (@var{mesh}, @var{beta})
##
## Build the UW stabilized stiffness matrix for an advection problem.
##
## The equation taken into account is:
##
## div (@var{beta} u) = f
##
## where @var{beta} is an element-wise constant vector function.
##
## Instead of passing the vector field @var{beta} directly one can pass
## a piecewise linear conforming scalar function @var{phi} as the last
## input. In such case @var{beta} = grad @var{phi} is assumed.
##
## If @var{phi} is a single scalar value @var{beta} is assumed to be 0
## in the whole domain.
##
## @seealso{bim2a_rhs, bim2a_reaction, bim2c_mesh_properties}
## @end deftypefn
function A = bim2a_advection_upwind (mesh, beta)
## Check input
if nargin != 2
error("bim2a_advection_upwind: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim2a_advection_upwind: first input is not a valid mesh structure.");
endif
nnodes = columns(mesh.p);
nelem = columns(mesh.t);
alphaareak = reshape (mesh.area, 1, 1, nelem);
shg = mesh.shg(:,:,:);
## Build local Laplacian matrix
Lloc = zeros(3,3,nelem);
for inode = 1:3
for jnode = 1:3
ginode(inode,jnode,:) = mesh.t(inode,:);
gjnode(inode,jnode,:) = mesh.t(jnode,:);
Lloc(inode,jnode,:) = sum( shg(:,inode,:) .* shg(:,jnode,:),1) .* alphaareak;
endfor
endfor
x = mesh.p(1,:);
x = x(mesh.t(1:3,:));
y = mesh.p(2,:);
y = y(mesh.t(1:3,:));
if all(size(beta)==1)
v12 = 0;
v23 = 0;
v31 = 0;
elseif all(size(beta)==[2,nelem])
v12 = beta(1,:) .* (x(2,:)-x(1,:)) + beta(2,:) .* (y(2,:)-y(1,:));
v23 = beta(1,:) .* (x(3,:)-x(2,:)) + beta(2,:) .* (y(3,:)-y(2,:));
v31 = beta(1,:) .* (x(1,:)-x(3,:)) + beta(2,:) .* (y(1,:)-y(3,:));
elseif all(size(beta)==[nnodes,1])
betaloc = beta(mesh.t(1:3,:));
v12 = betaloc(2,:)-betaloc(1,:);
v23 = betaloc(3,:)-betaloc(2,:);
v31 = betaloc(1,:)-betaloc(3,:);
else
error("bim2a_advection_upwind: coefficient beta has wrong dimensions.");
endif
[bp12, bm12] = deal (- (v12 - abs (v12))/2, (v12 + abs (v12))/2);
[bp23, bm23] = deal (- (v23 - abs (v23))/2, (v23 + abs (v23))/2);
[bp31, bm31] = deal (- (v31 - abs (v31))/2, (v31 + abs (v31))/2);
bp12 = reshape(bp12,1,1,nelem).*Lloc(1,2,:);
bm12 = reshape(bm12,1,1,nelem).*Lloc(1,2,:);
bp23 = reshape(bp23,1,1,nelem).*Lloc(2,3,:);
bm23 = reshape(bm23,1,1,nelem).*Lloc(2,3,:);
bp31 = reshape(bp31,1,1,nelem).*Lloc(3,1,:);
bm31 = reshape(bm31,1,1,nelem).*Lloc(3,1,:);
Sloc(1,1,:) = (-bm12-bp31);
Sloc(1,2,:) = bp12;
Sloc(1,3,:) = bm31;
Sloc(2,1,:) = bm12;
Sloc(2,2,:) = (-bp12-bm23);
Sloc(2,3,:) = bp23;
Sloc(3,1,:) = bp31;
Sloc(3,2,:) = bm23;
Sloc(3,3,:) = (-bm31-bp23);
A = sparse(ginode(:), gjnode(:), Sloc(:));
endfunction
bim/inst/bim2a_axisymmetric_advection_diffusion.m 000644 000765 000000 00000034066 12420212154 023112 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006-2014 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## author: Matteo porro
## author: Emanuela Abbate
## -*- texinfo -*-
## @deftypefn {Function File} @
## {[@var{A}]} = @
## bim2a_axisymmetric_advection_diffusion(@var{mesh},@var{alpha},@var{gamma},@var{eta},@var{beta})
##
## Build the Scharfetter-Gummel stabilized stiffness matrix for a
## diffusion-advection problem in cylindrical coordinates with axisymmetric
## configuration. Rotational symmetry is assumed with respect to be the vertical
## axis r=0. Only plane geometries that DO NOT intersect the symmetry axis
## are admitted.
##
##@example
##@group
## | ____ _|____
## | | \ \ | |
## z | | \ OK \| | NO!
## | |______\ |\___|
## | r |
#@end group
#@end example
##
## The equation taken into account is:
##
## 1/r * d(r * Fr)/dr + dFz/dz = f
##
## with
##
## F = [Fr, Fz]' = - @var{alpha} * @var{gamma} ( @var{eta} grad (u) - @var{beta} u )
##
## where @var{alpha} is an element-wise constant scalar function,
## @var{eta} and @var{gamma} are piecewise linear conforming scalar
## functions, @var{beta} is an element-wise constant vector function.
##
## Instead of passing the vector field @var{beta} directly, one can pass
## a piecewise linear conforming scalar function @var{phi} as the last
## input. In such case @var{beta} = grad @var{phi} is assumed.
##
## If @var{phi} is a single scalar value @var{beta} is assumed to be 0
## in the whole domain.
##
## @seealso{bim2a_axisymmetric_rhs, bim2a_axisymmetric_reaction,
## bim2a_advection_diffusion, bim2c_mesh_properties}
## @end deftypefn
function [A] = bim2a_axisymmetric_advection_diffusion (mesh, alpha, gamma, eta, beta)
## Check input
if nargin != 5
error("bim2a_axisymmetric_advection_diffusion: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim2a_axisymmetric_advection_diffusion: first input is not a valid mesh structure.");
elseif !(all(mesh.p(1,:) >= 0) || all(mesh.p(1,:) <= 0))
error("bim2a_axisymmetric_advection_diffusion: the input mesh cannot intersect the rotation axis r=0.");
endif
nnodes = columns(mesh.p);
nelem = columns(mesh.t);
## Turn scalar input to a vector of appropriate size
if isscalar(alpha)
alpha = alpha*ones(nelem,1);
endif
if isscalar(gamma)
gamma = gamma*ones(nnodes,1);
endif
if isscalar(eta)
eta = eta*ones(nnodes,1);
endif
if !( isvector(alpha) && isvector(gamma) && isvector(eta) )
error("bim2a_axisymmetric_advection_diffusion: coefficients are not valid vectors.");
elseif length(alpha) != nelem
error("bim2a_axisymmetric_advection_diffusion: length of alpha is not equal to the number of elements.");
elseif length(gamma) != nnodes
error("bim2a_axisymmetric_advection_diffusion: length of gamma is not equal to the number of nodes.");
elseif length(eta) != nnodes
error("bim2a_axisymmetric_advection_diffusion: length of eta is not equal to the number of nodes.");
endif
x = abs(mesh.p(1,:));
x = x(mesh.t(1:3,:));
y = mesh.p(2,:);
y = y(mesh.t(1:3,:));
alphaareak = reshape (alpha.*mesh.area,1,1,nelem);
shg = mesh.shg(:,:,:);
## Build local Laplacian matrix
Lloc = zeros(3,3,nelem);
for inode = 1:3
for jnode = 1:3
ginode(inode,jnode,:) = mesh.t(inode,:);
gjnode(inode,jnode,:) = mesh.t(jnode,:);
Lloc(inode,jnode,:) = sum( shg(:,inode,:) .* shg(:,jnode,:),1) .* alphaareak;
endfor
endfor
if all(size(beta)==1)
v12 = 0;
v23 = 0;
v31 = 0;
elseif all(size(beta)==[2,nelem])
v12 = beta(1,:) .* (x(2,:)-x(1,:)) + beta(2,:) .* (y(2,:)-y(1,:));
v23 = beta(1,:) .* (x(3,:)-x(2,:)) + beta(2,:) .* (y(3,:)-y(2,:));
v31 = beta(1,:) .* (x(1,:)-x(3,:)) + beta(2,:) .* (y(1,:)-y(3,:));
elseif all(size(beta)==[nnodes,1])
betaloc = beta(mesh.t(1:3,:));
v12 = betaloc(2,:)-betaloc(1,:);
v23 = betaloc(3,:)-betaloc(2,:);
v31 = betaloc(1,:)-betaloc(3,:);
else
error("bim2a_axisymmetric_advection_diffusion: coefficient beta has wrong dimensions.");
endif
etaloc = eta(mesh.t(1:3,:));
eta12 = etaloc(2,:) - etaloc(1,:);
eta23 = etaloc(3,:) - etaloc(2,:);
eta31 = etaloc(1,:) - etaloc(3,:);
r12 = (x(2,:) + x(1,:)) / 2;
r23 = (x(3,:) + x(2,:)) / 2;
r31 = (x(1,:) + x(3,:)) / 2;
etalocm1 = bimu_logm(etaloc(2,:),etaloc(3,:));
etalocm2 = bimu_logm(etaloc(3,:),etaloc(1,:));
etalocm3 = bimu_logm(etaloc(1,:),etaloc(2,:));
gammaloc = gamma(mesh.t(1:3,:));
geloc = gammaloc.*etaloc;
gelocm1 = bimu_logm (geloc(2,:), geloc(3,:));
gelocm2 = bimu_logm (geloc(3,:), geloc(1,:));
gelocm3 = bimu_logm (geloc(1,:), geloc(2,:));
[bp12,bm12] = bimu_bernoulli ((v12 - eta12) ./ etalocm3);
[bp23,bm23] = bimu_bernoulli ((v23 - eta23) ./ etalocm1);
[bp31,bm31] = bimu_bernoulli ((v31 - eta31) ./ etalocm2);
bp12 = reshape(r12.*gelocm3.*etalocm3.*bp12,1,1,nelem).*Lloc(1,2,:);
bm12 = reshape(r12.*gelocm3.*etalocm3.*bm12,1,1,nelem).*Lloc(1,2,:);
bp23 = reshape(r23.*gelocm1.*etalocm1.*bp23,1,1,nelem).*Lloc(2,3,:);
bm23 = reshape(r23.*gelocm1.*etalocm1.*bm23,1,1,nelem).*Lloc(2,3,:);
bp31 = reshape(r31.*gelocm2.*etalocm2.*bp31,1,1,nelem).*Lloc(3,1,:);
bm31 = reshape(r31.*gelocm2.*etalocm2.*bm31,1,1,nelem).*Lloc(3,1,:);
Sloc(1,1,:) = (-bm12-bp31)./reshape(etaloc(1,:),1,1,nelem);
Sloc(1,2,:) = bp12./reshape(etaloc(2,:),1,1,nelem);
Sloc(1,3,:) = bm31./reshape(etaloc(3,:),1,1,nelem);
Sloc(2,1,:) = bm12./reshape(etaloc(1,:),1,1,nelem);
Sloc(2,2,:) = (-bp12-bm23)./reshape(etaloc(2,:),1,1,nelem);
Sloc(2,3,:) = bp23./reshape(etaloc(3,:),1,1,nelem);
Sloc(3,1,:) = bp31./reshape(etaloc(1,:),1,1,nelem);
Sloc(3,2,:) = bm23./reshape(etaloc(2,:),1,1,nelem);
Sloc(3,3,:) = (-bm31-bp23)./reshape(etaloc(3,:),1,1,nelem);
A = sparse(ginode(:),gjnode(:),Sloc(:));
endfunction
%!test
%! n = 3;
%! [mesh] = msh2m_structured_mesh(linspace(1,2,n+1),linspace(0,1,n+1),1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! uex = @(r,z) exp(r);
%! duexdr = @(r,z) uex(r,z);
%! d2uexdr2 = @(r,z) uex(r,z);
%! duexdz = @(r,z) 0*uex(r,z);
%! d2uexdz2 = @(r,z) 0*uex(r,z);
%! Dnodes = bim2c_unknowns_on_side(mesh,[2,4]);
%! Nnodes = columns(mesh.p);
%! Nelements = columns(mesh.t);
%! Varnodes = setdiff(1:Nnodes,Dnodes);
%! D = 1; vr = 1; vz = 0;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! beta = 1/D*[vr*ones(1,Nelements); vz*ones(1,Nelements)];
%! f = @(r,z) -D./r.*duexdr(r,z) - D.*d2uexdr2(r,z) ...
%! + vr./r .* uex(r,z) + vr * duexdr(r,z) ...
%! - D.*d2uexdz2(r,z) + vz * duexdz(r,z);
%! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh.p(1,:), mesh.p(2,:)));
%! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1); u(Dnodes) = uex(mesh.p(1,Dnodes), mesh.p(2,Dnodes));
%! u(Varnodes) = S(Varnodes,Varnodes)\(rhs(Varnodes) - S(Varnodes,Dnodes)*u(Dnodes));
%! assert(u,uex(mesh.p(1,:), mesh.p(2,:))',1e-7)
%!test
%! n = 20;
%! [mesh] = msh2m_structured_mesh(linspace(1,2,n+1),linspace(0,1,n+1),1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! uex = @(r,z) exp(r) .* exp(1-z);
%! duexdr = @(r,z) uex(r,z);
%! d2uexdr2 = @(r,z) uex(r,z);
%! duexdz = @(r,z) -uex(r,z);
%! d2uexdz2 = @(r,z) uex(r,z);
%! Dnodes = bim2c_unknowns_on_side(mesh,[1,2,3,4]);
%! Nnodes = columns(mesh.p);
%! Nelements = columns(mesh.t);
%! Varnodes = setdiff(1:Nnodes,Dnodes);
%! D = 1; vr = 1; vz = 1;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! beta = 1/D*[vr*ones(1,Nelements); vz*ones(1,Nelements)];
%! f = @(r,z) -D./r.*duexdr(r,z) - D.*d2uexdr2(r,z) ...
%! + vr./r .* uex(r,z) + vr * duexdr(r,z) ...
%! - D.*d2uexdz2(r,z) + vz * duexdz(r,z);
%! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh.p(1,:), mesh.p(2,:)));
%! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1); u(Dnodes) = uex(mesh.p(1,Dnodes), mesh.p(2,Dnodes));
%! u(Varnodes) = S(Varnodes,Varnodes)\(rhs(Varnodes) - S(Varnodes,Dnodes)*u(Dnodes));
%! assert(u,uex(mesh.p(1,:), mesh.p(2,:))',1e-3)
%!test
%! n = 10;
%! [mesh] = msh2m_structured_mesh(linspace(1,2,n+1),linspace(0,1,n+1),1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! uex = @(r,z) exp(r) .* exp(1-z);
%! duexdr = @(r,z) uex(r,z);
%! d2uexdr2 = @(r,z) uex(r,z);
%! duexdz = @(r,z) -uex(r,z);
%! d2uexdz2 = @(r,z) uex(r,z);
%! Dnodes = bim2c_unknowns_on_side(mesh,[1,2,3,4]);
%! Nnodes = columns(mesh.p);
%! Nelements = columns(mesh.t);
%! Varnodes = setdiff(1:Nnodes,Dnodes);
%! D = 1;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! beta = 1/D * mesh.p(1,:)';
%! f = @(r,z) -D./r.*duexdr(r,z) - D.*d2uexdr2(r,z) ...
%! + 1./r .* uex(r,z) + duexdr(r,z) ...
%! - D.*d2uexdz2(r,z);
%! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh.p(1,:), mesh.p(2,:)));
%! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1); u(Dnodes) = uex(mesh.p(1,Dnodes), mesh.p(2,Dnodes));
%! u(Varnodes) = S(Varnodes,Varnodes)\(rhs(Varnodes) - S(Varnodes,Dnodes)*u(Dnodes));
%! assert(u,uex(mesh.p(1,:), mesh.p(2,:))',1e-3)
%!test
%! n = 10;
%! [mesh] = msh2m_structured_mesh(linspace(1,2,n+1),linspace(0,1,n+1),1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! uex = @(r,z) exp(r) .* exp(1-z);
%! duexdr = @(r,z) uex(r,z);
%! d2uexdr2 = @(r,z) uex(r,z);
%! duexdz = @(r,z) -uex(r,z);
%! d2uexdz2 = @(r,z) uex(r,z);
%! Dnodes = bim2c_unknowns_on_side(mesh,[1,2,3,4]);
%! Nnodes = columns(mesh.p);
%! Nelements = columns(mesh.t);
%! Varnodes = setdiff(1:Nnodes,Dnodes);
%! D = 1;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! beta = 1/D * 1/2*(mesh.p(1,:)').^2;
%! f = @(r,z) 1./r.*(1+r).*(r-D) .* uex(r,z);
%! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh.p(1,:), mesh.p(2,:)));
%! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1); u(Dnodes) = uex(mesh.p(1,Dnodes), mesh.p(2,Dnodes));
%! u(Varnodes) = S(Varnodes,Varnodes)\(rhs(Varnodes) - S(Varnodes,Dnodes)*u(Dnodes));
%! assert(u,uex(mesh.p(1,:), mesh.p(2,:))',1e-3)
%!test
%! n = 3;
%! [mesh] = msh2m_structured_mesh(linspace(-2,-1,n+1),linspace(0,1,n+1),1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! uex = @(r,z) exp(r);
%! duexdr = @(r,z) uex(r,z);
%! d2uexdr2 = @(r,z) uex(r,z);
%! duexdz = @(r,z) 0*uex(r,z);
%! d2uexdz2 = @(r,z) 0*uex(r,z);
%! Dnodes = bim2c_unknowns_on_side(mesh,[2,4]);
%! Nnodes = columns(mesh.p);
%! Nelements = columns(mesh.t);
%! Varnodes = setdiff(1:Nnodes,Dnodes);
%! D = 1; vr = 1; vz = 0;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! beta = 1/D*[vr*ones(1,Nelements); vz*ones(1,Nelements)];
%! f = @(r,z) -D./r.*duexdr(r,z) - D.*d2uexdr2(r,z) ...
%! + vr./r .* uex(r,z) + vr * duexdr(r,z) ...
%! - D.*d2uexdz2(r,z) + vz * duexdz(r,z);
%! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(abs(mesh.p(1,:)), mesh.p(2,:)));
%! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1); u(Dnodes) = uex(abs(mesh.p(1,Dnodes)), mesh.p(2,Dnodes));
%! u(Varnodes) = S(Varnodes,Varnodes)\(rhs(Varnodes) - S(Varnodes,Dnodes)*u(Dnodes));
%! assert(u,uex(abs(mesh.p(1,:)), mesh.p(2,:))',1e-7)
%!test
%! n = 10;
%! [mesh] = msh2m_structured_mesh(linspace(-2,-1,n+1),linspace(0,1,n+1),1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! uex = @(r,z) exp(r) .* exp(1-z);
%! duexdr = @(r,z) uex(r,z);
%! d2uexdr2 = @(r,z) uex(r,z);
%! duexdz = @(r,z) -uex(r,z);
%! d2uexdz2 = @(r,z) uex(r,z);
%! Dnodes = bim2c_unknowns_on_side(mesh,[1,2,3,4]);
%! Nnodes = columns(mesh.p);
%! Nelements = columns(mesh.t);
%! Varnodes = setdiff(1:Nnodes,Dnodes);
%! D = 1;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! beta = 1/D * 1/2*(mesh.p(1,:)').^2;
%! f = @(r,z) 1./r.*(1+r).*(r-D) .* uex(r,z);
%! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(abs(mesh.p(1,:)), mesh.p(2,:)));
%! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! u = zeros(Nnodes,1); u(Dnodes) = uex(abs(mesh.p(1,Dnodes)), mesh.p(2,Dnodes));
%! u(Varnodes) = S(Varnodes,Varnodes)\(rhs(Varnodes) - S(Varnodes,Dnodes)*u(Dnodes));
%! assert(u,uex(abs(mesh.p(1,:)), mesh.p(2,:))',1e-3)
%!test
%! [mesh] = msh2m_structured_mesh([0:.1:1],[0:.1:1],1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! x = mesh.p(1,:)'; y = mesh.p(2,:)';
%! Dnodes = bim2c_unknowns_on_side(mesh,[1:4]);
%! Nnodes = columns(mesh.p); Nelements = columns(mesh.t);
%! alpha = ones(Nelements,1); eta=ones(Nnodes,1);
%! beta = 0;
%! gamma = ones(Nnodes,1);
%! A = bim2a_axisymmetric_advection_diffusion(mesh,1,1,1,0);
%! B = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! assert(A,B)
bim/inst/bim2a_axisymmetric_advection_upwind.m 000644 000765 000000 00000010760 12420212154 022425 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006-2014 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## author: Matteo porro
## author: Emanuela Abbate
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {[@var{A}]} = @
## bim2a_axisymmetric_advection_upwind (@var{mesh}, @var{beta})
##
## Build the Upwind stabilized stiffness matrix for an advection problem
## in cylindrical coordinates with axisymmetric configuration.
##
## The equation taken into account is:
##
## 1/r * d/dr (r * @var{beta}_r u) + d/dz (@var{beta}_z u) = f
##
## where @var{beta} is an element-wise constant vector function.
##
## Instead of passing the vector field @var{beta} directly one can pass
## a piecewise linear conforming scalar function @var{phi} as the last
## input. In such case @var{beta} = grad @var{phi} is assumed.
##
## If @var{phi} is a single scalar value @var{beta} is assumed to be 0
## in the whole domain.
##
## @seealso{bim2a_axisymmetric_rhs, bim2a_axisymmetric_reaction,
## bim2a_axisymmetric_advection_diffusion, bim2c_mesh_properties}
## @end deftypefn
function A = bim2a_axisymmetric_advection_upwind (mesh, beta)
## Check input
if nargin != 2
error("bim2a_axisymmetric_advection_upwind: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim2a_axisymmetric_advection_upwind: first input is not a valid mesh structure.");
elseif !(all(mesh.p(1,:) >= 0) || all(mesh.p(1,:) <= 0))
error("bim2a_axisymmetric_advection_upwind: the input mesh cannot intersect the rotation axis r=0.");
endif
nnodes = columns(mesh.p);
nelem = columns(mesh.t);
x = abs (mesh.p(1,:));
x = x(mesh.t(1:3,:));
y = mesh.p(2,:);
y = y(mesh.t(1:3,:));
alphaareak = reshape (mesh.area, 1, 1, nelem);
shg = mesh.shg(:,:,:);
## Build local Laplacian matrix
Lloc = zeros(3,3,nelem);
for inode = 1:3
for jnode = 1:3
ginode(inode,jnode,:) = mesh.t(inode,:);
gjnode(inode,jnode,:) = mesh.t(jnode,:);
Lloc(inode,jnode,:) = sum( shg(:,inode,:) .* shg(:,jnode,:),1) .* alphaareak;
endfor
endfor
if all(size(beta)==1)
v12 = 0;
v23 = 0;
v31 = 0;
elseif all(size(beta)==[2,nelem])
v12 = beta(1,:) .* (x(2,:)-x(1,:)) + beta(2,:) .* (y(2,:)-y(1,:));
v23 = beta(1,:) .* (x(3,:)-x(2,:)) + beta(2,:) .* (y(3,:)-y(2,:));
v31 = beta(1,:) .* (x(1,:)-x(3,:)) + beta(2,:) .* (y(1,:)-y(3,:));
elseif all(size(beta)==[nnodes,1])
betaloc = beta(mesh.t(1:3,:));
v12 = betaloc(2,:)-betaloc(1,:);
v23 = betaloc(3,:)-betaloc(2,:);
v31 = betaloc(1,:)-betaloc(3,:);
else
error("bim2a_axisymmetric_advection_upwind: coefficient beta has wrong dimensions.");
endif
[bp12, bm12] = deal (- (v12 - abs (v12))/2, (v12 + abs (v12))/2);
[bp23, bm23] = deal (- (v23 - abs (v23))/2, (v23 + abs (v23))/2);
[bp31, bm31] = deal (- (v31 - abs (v31))/2, (v31 + abs (v31))/2);
r12 = (x(2,:) + x(1,:)) / 2;
r23 = (x(3,:) + x(2,:)) / 2;
r31 = (x(1,:) + x(3,:)) / 2;
bp12 = reshape(r12 .* bp12,1,1,nelem).*Lloc(1,2,:);
bm12 = reshape(r12 .* bm12,1,1,nelem).*Lloc(1,2,:);
bp23 = reshape(r23 .* bp23,1,1,nelem).*Lloc(2,3,:);
bm23 = reshape(r23 .* bm23,1,1,nelem).*Lloc(2,3,:);
bp31 = reshape(r31 .* bp31,1,1,nelem).*Lloc(3,1,:);
bm31 = reshape(r31 .* bm31,1,1,nelem).*Lloc(3,1,:);
Sloc(1,1,:) = (-bm12-bp31);
Sloc(1,2,:) = bp12;
Sloc(1,3,:) = bm31;
Sloc(2,1,:) = bm12;
Sloc(2,2,:) = (-bp12-bm23);
Sloc(2,3,:) = bp23;
Sloc(3,1,:) = bp31;
Sloc(3,2,:) = bm23;
Sloc(3,3,:) = (-bm31-bp23);
A = sparse(ginode(:), gjnode(:), Sloc(:));
endfunction
bim/inst/bim2a_axisymmetric_boundary_mass.m 000644 000765 000000 00000014473 12316045351 021745 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006-2014 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## author: Matteo porro
## author: Emanuela Abbate
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{M}]} = @
## bim2a_axisymmetric_boundary_mass(@var{mesh},@var{sidelist},@var{nodelist})
##
## Build the lumped boundary mass matrix needed to apply Robin and Neumann
## boundary conditions in a problem in cylindrical coordinates with
## axisymmetric configuration.
##
## The vector @var{sidelist} contains the list of the side edges
## contributing to the mass matrix.
##
## The optional argument @var{nodelist} contains the list of the
## degrees of freedom on the boundary.
##
## @seealso{bim2a_axisymmetric_rhs, bim2a_axisymmetric_advection_diffusion,
## bim2a_axisymmetric_laplacian, bim2a_axisymmetric_reaction, bim2a_boundary_mass}
## @end deftypefn
function [M] = bim2a_axisymmetric_boundary_mass(mesh,sidelist,nodelist)
## Check input
if (nargin > 3)
error ("bim2a_axisymmetric_boundary_mass: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim2a_axisymmetric_boundary_mass: first input is not a valid mesh structure.");
elseif !( isvector(sidelist) && isnumeric(sidelist) )
error("bim2a_axisymmetric_boundary_mass: second input is not a valid numeric vector.");
elseif !(all(mesh.p(1,:) >= 0) || all(mesh.p(1,:) <= 0))
error("bim2a_axisymmetric_boundary_mass: the input mesh cannot intersect the rotation axis r=0.");
endif
if (nargin < 3)
[nodelist] = bim2c_unknowns_on_side(mesh,sidelist);
endif
r = abs (mesh.p(1,nodelist));
edges = [];
for ie = sidelist
edges = [ edges, mesh.e([1:2 5],mesh.e(5,:)==ie)];
endfor
l = sqrt((mesh.p(1,edges(1,:))-mesh.p(1,edges(2,:))).^2 +
(mesh.p(2,edges(1,:))-mesh.p(2,edges(2,:))).^2);
dd = zeros(size(nodelist));
for in = 1:length(nodelist)
dd (in) = ( sum(r(in).*l(edges(1,:)==nodelist(in))) ...
+ sum(r(in).*l(edges(2,:)==nodelist(in))) )/2;
endfor
M = sparse(diag(dd));
endfunction
%!test
%! n = 3;
%! [mesh] = msh2m_structured_mesh(linspace(1,2,n+1),linspace(0,1,n+1),1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! uex = @(r,z) exp(r);
%! duexdr = @(r,z) uex(r,z);
%! d2uexdr2 = @(r,z) uex(r,z);
%! duexdz = @(r,z) 0*uex(r,z);
%! d2uexdz2 = @(r,z) 0*uex(r,z);
%! Rnodesr = bim2c_unknowns_on_side(mesh,[2]);
%! Rnodesl = bim2c_unknowns_on_side(mesh,[4]);
%! Nnodes = columns(mesh.p);
%! Nelements = columns(mesh.t);
%! D = 1; vr = 1; vz = 0;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! beta = 1/D*[vr*ones(1,Nelements); vz*ones(1,Nelements)];
%! f = @(r,z) -D./r.*duexdr(r,z) - D.*d2uexdr2(r,z) ...
%! + vr./r .* uex(r,z) + vr * duexdr(r,z) ...
%! - D.*d2uexdz2(r,z) + vz * duexdz(r,z);
%! gr = @(r,z) uex(r,z) - 1 * (-D*duexdr(r,z) + vr*uex(r,z));
%! gl = @(r,z) uex(r,z) - (-1) * (-D*duexdr(r,z) + vr*uex(r,z));
%! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh.p(1,:), mesh.p(2,:)));
%! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! Mr = bim2a_axisymmetric_boundary_mass(mesh,2); Ml = bim2a_axisymmetric_boundary_mass(mesh,4);
%! S(Rnodesr,Rnodesr) += Mr;
%! rhs(Rnodesr) += diag(Mr) .* gr(mesh.p(1,Rnodesr), mesh.p(2,Rnodesr))';
%! S(Rnodesl,Rnodesl) += Ml;
%! rhs(Rnodesl) += diag(Ml) .* gl(mesh.p(1,Rnodesl), mesh.p(2,Rnodesl))';
%! u = S\rhs;
%! assert(u,uex(mesh.p(1,:), mesh.p(2,:))',1e-7)
%!test
%! n = 10;
%! [mesh] = msh2m_structured_mesh(linspace(1,2,n+1),linspace(0,1,n+1),1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! uex = @(r,z) exp(r);
%! duexdr = @(r,z) uex(r,z);
%! d2uexdr2 = @(r,z) uex(r,z);
%! duexdz = @(r,z) 0*uex(r,z);
%! d2uexdz2 = @(r,z) 0*uex(r,z);
%! Rnodesr = bim2c_unknowns_on_side(mesh,[2]);
%! Rnodesl = bim2c_unknowns_on_side(mesh,[4]);
%! Rnodesb = bim2c_unknowns_on_side(mesh,[1]);
%! Rnodest = bim2c_unknowns_on_side(mesh,[3]);
%! Nnodes = columns(mesh.p);
%! Nelements = columns(mesh.t);
%! D = 1; vr = 1; vz = 0;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! beta = 1/D*[vr*ones(1,Nelements); vz*ones(1,Nelements)];
%! f = @(r,z) -D./r.*duexdr(r,z) - D.*d2uexdr2(r,z) ...
%! + vr./r .* uex(r,z) + vr * duexdr(r,z) ...
%! - D.*d2uexdz2(r,z) + vz * duexdz(r,z);
%! gr = @(r,z) uex(r,z) - 1 * (-D*duexdr(r,z) + vr*uex(r,z));
%! gl = @(r,z) uex(r,z) - (-1) * (-D*duexdr(r,z) + vr*uex(r,z));
%! gb = @(r,z) uex(r,z) - (-1) * (-D*duexdz(r,z) + vz*uex(r,z));
%! gt = @(r,z) uex(r,z) - 1 * (-D*duexdz(r,z) + vz*uex(r,z));
%! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh.p(1,:), mesh.p(2,:)));
%! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! Mr = bim2a_axisymmetric_boundary_mass(mesh,2); Ml = bim2a_axisymmetric_boundary_mass(mesh,4);
%! Mb = bim2a_axisymmetric_boundary_mass(mesh,1); Mt = bim2a_axisymmetric_boundary_mass(mesh,3);
%! S(Rnodesr,Rnodesr) += Mr;
%! rhs(Rnodesr) += diag(Mr) .* gr(mesh.p(1,Rnodesr), mesh.p(2,Rnodesr))';
%! S(Rnodesl,Rnodesl) += Ml;
%! rhs(Rnodesl) += diag(Ml) .* gl(mesh.p(1,Rnodesl), mesh.p(2,Rnodesl))';
%! S(Rnodesb,Rnodesb) += Mb;
%! rhs(Rnodesb) += diag(Mb) .* gb(mesh.p(1,Rnodesb), mesh.p(2,Rnodesb))';
%! S(Rnodest,Rnodest) += Mt;
%! rhs(Rnodest) += diag(Mt) .* gt(mesh.p(1,Rnodest), mesh.p(2,Rnodest))';
%! u = S\rhs;
%! assert(u,uex(mesh.p(1,:), mesh.p(2,:))',1e-7)
bim/inst/bim2a_axisymmetric_laplacian.m 000644 000765 000000 00000005324 12316045351 021016 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006-2014 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## author: Matteo porro
## author: Emanuela Abbate
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {@var{A}} = bim2a_axisymmetric_laplacian (@var{mesh},@var{epsilon},@var{kappa})
##
## Build the standard finite element stiffness matrix for a diffusion
## problem in cylindrical coordinates with axisymmetric configuration.
## Rotational symmetry is assumed with respect to be the vertical axis r=0.
## Only plane geometries that DO NOT intersect the symmetry axis are admitted.
##
##@example
##@group
## | ____ _|____
## | | \ \ | |
## z | | \ OK \| | NO!
## | |______\ |\___|
## | r |
#@end group
#@end example
##
## The equation taken into account is:
##
## 1/r * d(r * Fr)/dr + dFz/dz = f
##
## with
##
## F = [Fr, Fz]' = - @var{epsilon} * @var{kappa} grad (u)
##
## where @var{epsilon} is an element-wise constant scalar function,
## while @var{kappa} is a piecewise linear conforming scalar function.
##
## @seealso{bim2a_axisymmetric_rhs, bim2a_axisymmetric_reaction,
## bim2a_axisymmetric_advection_diffusion, bim2a_laplacian, bim1a_laplacian,
## bim3a_laplacian}
## @end deftypefn
function [A] = bim2a_axisymmetric_laplacian(mesh,epsilon,kappa)
## Check input
if nargin != 3
error("bim2a_axisymmetric_laplacian: wrong number of input parameters.");
elseif !(all(mesh.p(1,:) >= 0) || all(mesh.p(1,:) <= 0))
error("bim2a_axisymmetric_laplacian: the input mesh cannot intersect the rotation axis r=0.");
endif
## Input check inside bim2a_axisymmetric_advection_diffusion
nnodes = columns(mesh.p);
nelem = columns(mesh.t);
A = bim2a_axisymmetric_advection_diffusion (mesh,epsilon,kappa,ones(nnodes,1),0);
endfunction
bim/inst/bim2a_axisymmetric_reaction.m 000644 000765 000000 00000012202 12316045351 020667 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006-2014 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## author: Matteo porro
## author: Emanuela Abbate
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{C}]} = @
## bim2a_axisymmetric_reaction(@var{mesh},@var{delta},@var{zeta})
##
## Build the lumped finite element mass matrix for a diffusion
## problem in cylindrical coordinates with axisymmetric configuration.
##
## The equation taken into account is:
##
## @var{delta} * @var{zeta} * u = f
##
## where @var{delta} is an element-wise constant scalar function, while
## @var{zeta} is a piecewise linear conforming scalar function.
##
## @seealso{bim2a_rhs, bim2a_axisymmetric_advection_diffusion,
## bim2a_axisymmetric_laplacian, bim2a_reaction, bim1a_reaction, bim3a_reaction}
## @end deftypefn
function [C] = bim2a_axisymmetric_reaction(mesh,delta,zeta)
## Check input
if nargin != 3
error("bim2a_axisymmetric_reaction: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim2a_axisymmetric_reaction: first input is not a valid mesh structure.");
elseif !(all(mesh.p(1,:) >= 0) || all(mesh.p(1,:) <= 0))
error("bim2a_axisymmetric_reaction: the input mesh cannot intersect the rotation axis r=0.");
endif
nnodes = size(mesh.p,2);
nelem = size(mesh.t,2);
r = abs (mesh.p(1,:));
## Turn scalar input to a vector of appropriate size
if isscalar(delta)
delta = delta*ones(nelem,1);
endif
if isscalar(zeta)
zeta = zeta*ones(nnodes,1);
endif
if !( isvector(delta) && isvector(zeta) )
error("bim2a_axisymmetric_reaction: coefficients are not valid vectors.");
elseif length(delta) != nelem
error("bim2a_axisymmetric_reaction: length of alpha is not equal to the number of elements.");
elseif length(zeta) != nnodes
error("bim2a_axisymmetric_reaction: length of gamma is not equal to the number of nodes.");
endif
wjacdet = mesh.wjacdet(:,:);
coeff = zeta(mesh.t(1:3,:));
coeffe = delta(:);
## Local matrix
Blocmat = zeros(3,nelem);
for inode = 1:3
Blocmat(inode,:) = coeffe'.*coeff(inode,:).*wjacdet(inode,:) .* r(mesh.t(inode,:));
endfor
gnode = (mesh.t(1:3,:));
## Global matrix
C = sparse(gnode(:),gnode(:),Blocmat(:));
endfunction
%!shared mesh,delta,zeta,nnodes,nelem
% x = y = linspace(0,1,4);
% [mesh] = msh2m_structured_mesh(x,y,1,1:4);
% [mesh] = bim2c_mesh_properties(mesh);
% nnodes = columns(mesh.p);
% nelem = columns(mesh.t);
% delta = ones(columns(mesh.t),1);
% zeta = ones(columns(mesh.p),1);
%!test
% [C] = bim2a_axisymmetric_reaction(mesh,delta,zeta);
% assert(size(C),[nnodes, nnodes]);
%!test
% [C1] = bim2a_axisymmetric_reaction(mesh,3*delta,zeta);
% [C2] = bim2a_axisymmetric_reaction(mesh,delta,3*zeta);
% assert(C1,C2);
%!test
% [C1] = bim2a_axisymmetric_reaction(mesh,3*delta,zeta);
% [C2] = bim2a_axisymmetric_reaction(mesh,3,1);
% assert(C1,C2);
%!test
%! n = 20;
%! [mesh] = msh2m_structured_mesh(linspace(1,2,n+1),linspace(0,1,n+1),1,1:4);
%! mesh = bim2c_mesh_properties(mesh);
%! uex = @(r,z) exp(r) .* exp(1-z);
%! duexdr = @(r,z) uex(r,z);
%! d2uexdr2 = @(r,z) uex(r,z);
%! duexdz = @(r,z) -uex(r,z);
%! d2uexdz2 = @(r,z) uex(r,z);
%! Dnodes = bim2c_unknowns_on_side(mesh,[1,2,3,4]);
%! Nnodes = columns(mesh.p);
%! Nelements = columns(mesh.t);
%! Varnodes = setdiff(1:Nnodes,Dnodes);
%! D = 1; vr = 1; vz = 1; sigma = 1;
%! alpha = D*ones(Nelements,1);
%! gamma = ones(Nnodes,1);
%! eta = ones(Nnodes,1);
%! delta = sigma*ones(columns(mesh.t),1);
%! zeta = ones(columns(mesh.p),1);
%! beta = 1/D*[vr*ones(1,Nelements); vz*ones(1,Nelements)];
%! f = @(r,z) -D./r.*duexdr(r,z) - D.*d2uexdr2(r,z) ...
%! + vr./r .* uex(r,z) + vr * duexdr(r,z) ...
%! - D.*d2uexdz2(r,z) + vz * duexdz(r,z) ...
%! + sigma * uex(r,z);
%! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh.p(1,:), mesh.p(2,:)));
%! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta);
%! C = bim2a_axisymmetric_reaction(mesh,delta,zeta);
%! S += C;
%! u = zeros(Nnodes,1); u(Dnodes) = uex(mesh.p(1,Dnodes), mesh.p(2,Dnodes));
%! u(Varnodes) = S(Varnodes,Varnodes)\(rhs(Varnodes) - S(Varnodes,Dnodes)*u(Dnodes));
%! assert(u,uex(mesh.p(1,:), mesh.p(2,:))',1e-3)
bim/inst/bim2a_axisymmetric_rhs.m 000644 000765 000000 00000007144 12420212155 017662 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006-2014 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## author: Matteo porro
## author: Emanuela Abbate
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{b}]} = @
## bim2a_axisymmetric_rhs(@var{mesh},@var{f},@var{g})
##
## Build the finite element right-hand side of a diffusion problem
## in cylindrical coordinates with axisymmetric configuration
## employing mass-lumping.
##
## The equation taken into account is:
##
## @var{delta} * u = f * g
##
## where @var{f} is an element-wise constant scalar function, while
## @var{g} is a piecewise linear conforming scalar function.
##
## @seealso{bim2a_axisymmetric_reaction, bim2a_axisymmetric_advection_diffusion,
## bim2a_axisymmetric_laplacian, bim1a_axisymmetric_rhs}
## @end deftypefn
function b = bim2a_axisymmetric_rhs(mesh,f,g)
## Check input
if (nargin != 3)
error("bim2a_axisymmetric_rhs: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim2a_axisymmetric_rhs: first input is not a valid mesh structure.");
elseif !(all(mesh.p(1,:) >= 0) || all(mesh.p(1,:) <= 0))
error("bim2a_axisymmetric_rhs: the input mesh cannot intersect the rotation axis r=0.");
endif
nnodes = columns(mesh.p);
nelem = columns(mesh.t);
r = abs (mesh.p(1,:));
## Turn scalar input to a vector of appropriate size
if isscalar(f)
f = f*ones(nelem,1);
endif
if isscalar(g)
g = g*ones(nnodes,1);
endif
if !( isvector(f) && isvector(g) )
error("bim2a_axisymmetric_rhs: coefficients are not valid vectors.");
elseif length(f) != nelem
error("bim2a_axisymmetric_rhs: length of f is not equal to the number of elements.");
elseif length(g) != nnodes
error("bim2a_axisymmetric_rhs: length of g is not equal to the number of nodes.");
endif
g = g(mesh.t(1:3,:));
wjacdet = mesh.wjacdet;
## Build local matrix
Blocmat = zeros(3,nelem);
for inode = 1:3
Blocmat(inode,:) = f'.*g(inode,:).*wjacdet(inode,:) .* r(mesh.t(inode,:));
endfor
gnode = (mesh.t(1:3,:));
## Assemble global matrix
b = sparse(gnode(:),1,Blocmat(:));
endfunction
%!shared mesh,f,g,nnodes,nelem
% x = y = linspace(0,1,4);
% [mesh] = msh2m_structured_mesh(x,y,1,1:4);
% [mesh] = bim2c_mesh_properties(mesh);
% nnodes = columns(mesh.p);
% nelem = columns(mesh.t);
% g = ones(columns(mesh.t),1);
% f = ones(columns(mesh.p),1);
%!test
% [b] = bim2a_axisymmetric_rhs(mesh,f,g);
% assert(size(b),[nnodes, 1]);
%!test
% [b1] = bim2a_axisymmetric_rhs(mesh,3*f,g);
% [b2] = bim2a_axisymmetric_rhs(mesh,f,3*g);
% assert(b1,b2);
%!test
% [b1] = bim2a_axisymmetric_rhs(mesh,3*f,g);
% [b2] = bim2a_axisymmetric_rhs(mesh,3,1);
% assert(b1,b2);
bim/inst/bim2a_boundary_mass.m 000644 000765 000000 00000004613 12316041364 017142 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{M}]} = @
## bim2a_boundary_mass(@var{mesh},@var{sidelist},@var{nodelist})
##
## Build the lumped boundary mass matrix needed to apply Robin boundary
## conditions.
##
## The vector @var{sidelist} contains the list of the side edges
## contributing to the mass matrix.
##
## The optional argument @var{nodelist} contains the list of the
## degrees of freedom on the boundary.
##
## @seealso{bim2a_rhs, bim2a_advection_diffusion, bim2a_laplacian,
## bim2a_reaction}
## @end deftypefn
function [M] = bim2a_boundary_mass(mesh,sidelist,nodelist)
## Check input
if nargin > 3
error("bim2a_boundary_mass: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim2a_boundary_mass: first input is not a valid mesh structure.");
elseif !( isvector(sidelist) && isnumeric(sidelist) )
error("bim2a_boundary_mass: second input is not a valid numeric vector.");
endif
if nargin<3
[nodelist] = bim2c_unknowns_on_side(mesh,sidelist);
endif
edges = [];
for ie = sidelist
edges = [ edges, mesh.e([1:2 5],mesh.e(5,:)==ie)];
endfor
l = sqrt((mesh.p(1,edges(1,:))-mesh.p(1,edges(2,:))).^2 +
(mesh.p(2,edges(1,:))-mesh.p(2,edges(2,:))).^2);
dd = zeros(size(nodelist));
for in = 1:numel (nodelist)
dd (in) = (sum(l(edges(1,:)==nodelist(in)))+sum(l(edges(2,:)==nodelist(in))))/2;
endfor
M = sparse(diag(dd));
endfunction
bim/inst/bim2a_laplacian.m 000644 000765 000000 00000003470 12041000520 016200 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {@var{A}} = bim2a_laplacian (@var{mesh},@var{epsilon},@var{kappa})
##
## Build the standard finite element stiffness matrix for a diffusion
## problem.
##
## The equation taken into account is:
##
## - div (@var{epsilon} * @var{kappa} grad (u)) = f
##
## where @var{epsilon} is an element-wise constant scalar function,
## while @var{kappa} is a piecewise linear conforming scalar function.
##
## @seealso{bim2a_rhs, bim2a_reaction, bim2a_advection_diffusion, bim1a_laplacian, bim3a_laplacian}
## @end deftypefn
function [A] = bim2a_laplacian(mesh,epsilon,kappa)
## Check input
if nargin != 3
error("bim2a_laplacian: wrong number of input parameters.");
endif
## Input check inside bim2a_advection_diffusion
nnodes = columns(mesh.p);
nelem = columns(mesh.t);
A = bim2a_advection_diffusion (mesh,epsilon,kappa,ones(nnodes,1),0);
endfunction bim/inst/bim2a_reaction.m 000644 000765 000000 00000006330 12316041364 016076 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{C}]} = @
## bim2a_reaction(@var{mesh},@var{delta},@var{zeta})
##
## Build the lumped finite element mass matrix for a diffusion
## problem.
##
## The equation taken into account is:
##
## @var{delta} * @var{zeta} * u = f
##
## where @var{delta} is an element-wise constant scalar function, while
## @var{zeta} is a piecewise linear conforming scalar function.
##
## @seealso{bim2a_rhs, bim2a_advection_diffusion, bim2a_laplacian,
## bim1a_reaction, bim3a_reaction}
## @end deftypefn
function [C] = bim2a_reaction(mesh,delta,zeta)
## Check input
if nargin != 3
error("bim2a_reaction: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim2a_reaction: first input is not a valid mesh structure.");
endif
nnodes = size(mesh.p,2);
nelem = size(mesh.t,2);
## Turn scalar input to a vector of appropriate size
if isscalar(delta)
delta = delta*ones(nelem,1);
endif
if isscalar(zeta)
zeta = zeta*ones(nnodes,1);
endif
if !( isvector(delta) && isvector(zeta) )
error("bim2a_reaction: coefficients are not valid vectors.");
elseif (numel (delta) != nelem)
error("bim2a_: length of alpha is not equal to the number of elements.");
elseif (numel (zeta) != nnodes)
error("bim2a_: length of gamma is not equal to the number of nodes.");
endif
wjacdet = mesh.wjacdet(:,:);
coeff = zeta(mesh.t(1:3,:));
coeffe = delta(:);
## Local matrix
Blocmat = zeros(3,nelem);
for inode = 1:3
Blocmat(inode,:) = coeffe'.*coeff(inode,:).*wjacdet(inode,:);
endfor
gnode = (mesh.t(1:3,:));
## Global matrix
C = sparse(gnode(:),gnode(:),Blocmat(:));
endfunction
%!shared mesh,delta,zeta,nnodes,nelem
% x = y = linspace(0,1,4);
% [mesh] = msh2m_structured_mesh(x,y,,1:4;
% [mesh] = bim2c_mesh_properties(mesh);
% nnodes = columns(mesh.p);
% nelem = columns(mesh.t);
% delta = ones(columns(mesh.t),1);
% zeta = ones(columns(mesh.p),1);
%!test
% [C] = bim2a_reaction(mesh,delta,zeta);
% assert(size(C),[nnodes, nnodes]);
%!test
% [C1] = bim2a_reaction(mesh,3*delta,zeta);
% [C2] = bim2a_reaction(mesh,delta,3*zeta);
% assert(C1,C2);
%!test
% [C1] = bim2a_reaction(mesh,3*delta,zeta);
% [C2] = bim2a_reaction(mesh,3,1);
% assert(C1,C2);
bim/inst/bim2a_rhs.m 000644 000765 000000 00000006046 12316041364 015072 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{b}]} = @
## bim2a_rhs(@var{mesh},@var{f},@var{g})
##
## Build the finite element right-hand side of a diffusion problem
## employing mass-lumping.
##
## The equation taken into account is:
##
## @var{delta} * u = f * g
##
## where @var{f} is an element-wise constant scalar function, while
## @var{g} is a piecewise linear conforming scalar function.
##
## @seealso{bim2a_reaction, bim2a_advection_diffusion, bim2a_laplacian,
## bim1a_reaction, bim3a_reaction}
## @end deftypefn
function b = bim2a_rhs(mesh,f,g)
## Check input
if nargin != 3
error("bim2a_rhs: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim2a_rhs: first input is not a valid mesh structure.");
endif
nnodes = columns(mesh.p);
nelem = columns(mesh.t);
## Turn scalar input to a vector of appropriate size
if isscalar(f)
f = f*ones(nelem,1);
endif
if isscalar(g)
g = g*ones(nnodes,1);
endif
if !( isvector(f) && isvector(g) )
error("bim2a_rhs: coefficients are not valid vectors.");
elseif (numel (f) != nelem)
error("bim2a_rhs: length of f is not equal to the number of elements.");
elseif (numel (g) != nnodes)
error("bim2a_rhs: length of g is not equal to the number of nodes.");
endif
g = g(mesh.t(1:3,:));
wjacdet = mesh.wjacdet;
## Build local matrix
Blocmat=zeros(3,nelem);
for inode=1:3
Blocmat(inode,:) = f'.*g(inode,:).*wjacdet(inode,:);
endfor
gnode=(mesh.t(1:3,:));
## Assemble global matrix
b = sparse(gnode(:),1,Blocmat(:));
endfunction
%!shared mesh,f,g,nnodes,nelem
% x = y = linspace(0,1,4);
% [mesh] = msh2m_structured_mesh(x,y,1,1:4);
% [mesh] = bim2c_mesh_properties(mesh);
% nnodes = columns(mesh.p);
% nelem = columns(mesh.t);
% g = ones(columns(mesh.t),1);
% f = ones(columns(mesh.p),1);
%!test
% [b] = bim2a_rhs(mesh,f,g);
% assert(size(b),[nnodes, 1]);
%!test
% [b1] = bim2a_rhs(mesh,3*f,g);
% [b2] = bim2a_rhs(mesh,f,3*g);
% assert(b1,b2);
%!test
% [b1] = bim2a_rhs(mesh,3*f,g);
% [b2] = bim2a_rhs(mesh,3,1);
% assert(b1,b2);
bim/inst/bim2c_global_flux.m 000644 000765 000000 00000013453 12316041364 016576 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{jx},@var{jy}]} = @
## bim2c_global_flux(@var{mesh},@var{u},@var{alpha},@var{gamma},@var{eta},@var{beta})
##
## Compute the flux associated with the Scharfetter-Gummel approximation
## of the scalar field @var{u}.
##
## The vector field is defined as:
##
## J(@var{u}) = @var{alpha}* @var{gamma} * (@var{eta} * grad @var{u} - @var{beta} * @var{u}))
##
## where @var{alpha} is an element-wise constant scalar function,
## @var{eta} and @var{gamma} are piecewise linear conforming scalar
## functions, while @var{beta} is element-wise constant vector function.
##
## J(@var{u}) is an element-wise constant vector function.
##
## Instead of passing the vector field @var{beta} directly one can pass
## a piecewise linear conforming scalar function @var{phi} as the last
## input. In such case @var{beta} = grad @var{phi} is assumed. If
## @var{phi} is a single scalar value @var{beta} is assumed to be 0 in
## the whole domain.
##
## @seealso{bim2c_pde_gradient,bim2a_advection_diffusion}
## @end deftypefn
function [jx, jy] = bim2c_global_flux(mesh,u,alpha,gamma,eta,beta)
## Check input
if nargin != 6
error("bim2c_global_flux: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim2c_global_flux: first input is not a valid mesh structure.");
endif
nnodes = columns(mesh.p);
nelem = columns(mesh.t);
if !( isvector(u) && isvector(alpha) && isvector(gamma) && isvector(eta) )
error("bim2c_global_flux: coefficients are not valid vectors.");
elseif (numel (u) != nnodes)
error("bim2c_global_flux: length of u is not equal to the number of nodes.");
elseif (numel (alpha) != nelem)
error("bim2c_global_flux: length of alpha is not equal to the number of elements.");
elseif (numel (gamma) != nnodes)
error("bim2c_global_flux: length of gamma is not equal to the number of nodes.");
elseif (numel (eta) != nnodes)
error("bim2c_global_flux: length of eta is not equal to the number of nodes.");
endif
nelem = columns(mesh.t);
nnodes = columns(mesh.p);
uloc = u(mesh.t(1:3,:));
shgx = reshape(mesh.shg(1,:,:),3,nelem);
shgy = reshape(mesh.shg(2,:,:),3,nelem);
x = reshape(mesh.p(1,mesh.t(1:3,:)),3,[]);
dx = [ (x(3,:)-x(2,:)) ;
(x(1,:)-x(3,:)) ;
(x(2,:)-x(1,:)) ];
y = reshape(mesh.p(2,mesh.t(1:3,:)),3,[]);
dy = [ (y(3,:)-y(2,:)) ;
(y(1,:) -y(3,:)) ;
(y(2,:) -y(1,:)) ];
if all(size(beta)==1)
v12=0;v23=0;v31=0;
elseif all(size(beta)==[2,nelem])
v23 = beta(1,:) .* dx(1,:) + beta(2,:) .* dy(1,:);
v31 = beta(1,:) .* dx(2,:) + beta(2,:) .* dy(2,:);
v12 = beta(1,:) .* dx(3,:) + beta(2,:) .* dy(3,:);
elseif all(size(beta)==[nnodes,1])
betaloc = beta(mesh.t(1:3,:));
v23 = betaloc(3,:)-betaloc(2,:);
v31 = betaloc(1,:)-betaloc(3,:);
v12 = betaloc(2,:)-betaloc(1,:);
else
error("bim2c_global_flux: coefficient beta has wrong dimensions.");
endif
etaloc = eta(mesh.t(1:3,:));
eta23 = etaloc(3,:)-etaloc(2,:);
eta31 = etaloc(1,:)-etaloc(3,:);
eta12 = etaloc(2,:)-etaloc(1,:);
etalocm1 = bimu_logm(etaloc(2,:),etaloc(3,:));
etalocm2 = bimu_logm(etaloc(3,:),etaloc(1,:));
etalocm3 = bimu_logm(etaloc(1,:),etaloc(2,:));
gammaloc = gamma(mesh.t(1:3,:));
geloc = gammaloc.*etaloc;
gelocm1 = bimu_logm(geloc(2,:),geloc(3,:));
gelocm2 = bimu_logm(geloc(3,:),geloc(1,:));
gelocm3 = bimu_logm(geloc(1,:),geloc(2,:));
[bp23,bm23] = bimu_bernoulli( (v23 - eta23)./etalocm1);
[bp31,bm31] = bimu_bernoulli( (v31 - eta31)./etalocm2);
[bp12,bm12] = bimu_bernoulli( (v12 - eta12)./etalocm3);
gfigfj = [ shgx(3,:) .* shgx(2,:) + shgy(3,:) .* shgy(2,:) ;
shgx(1,:) .* shgx(3,:) + shgy(1,:) .* shgy(3,:) ;
shgx(2,:) .* shgx(1,:) + shgy(2,:) .* shgy(1,:) ];
jx = - alpha' .* ( gelocm1 .* etalocm1 .* dx(1,:) .* ...
gfigfj(1,:) .* ...
( bp23 .* uloc(3,:)./etaloc(3,:) -...
bm23 .* uloc(2,:)./etaloc(2,:)) +... %% 1
gelocm2 .* etalocm2 .* dx(2,:) .* ...
gfigfj(2,:) .* ...
(bp31 .* uloc(1,:)./etaloc(1,:) -...
bm31 .* uloc(3,:)./etaloc(3,:)) +... %% 2
gelocm3 .* etalocm3 .* dx(3,:) .* ...
gfigfj(3,:) .* ...
(bp12 .* uloc(2,:)./etaloc(2,:) -...
bm12 .* uloc(1,:)./etaloc(1,:)) ... %% 3
);
jy = - alpha' .* ( gelocm1 .* etalocm1 .* dy(1,:) .* ...
gfigfj(1,:) .* ...
( bp23 .* uloc(3,:)./etaloc(3,:) -...
bm23 .* uloc(2,:)./etaloc(2,:)) +... %% 1
gelocm2 .* etalocm2 .* dy(2,:) .* ...
gfigfj(2,:) .* ...
(bp31 .* uloc(1,:)./etaloc(1,:) -...
bm31 .* uloc(3,:)./etaloc(3,:)) +... %% 2
gelocm3 .* etalocm3 .* dy(3,:) .* ...
gfigfj(3,:) .* ...
(bp12 .* uloc(2,:)./etaloc(2,:) -...
bm12 .* uloc(1,:)./etaloc(1,:)) ... %% 3
);
endfunction
bim/inst/bim2c_intrp.m 000644 000765 000000 00000004161 12420212155 015422 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2011, 2012 Carlo de Falco
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## .
## -*- texinfo -*-
##
## @deftypefn {Function File} {@var{data}} = bim2c_intrp (@var{msh}, @var{n_data}, @var{e_data}, @var{points})
##
## Compute interpolated values of multicomponent node centered field @var{n_data} and/or
## cell centered field @var{n_data} at an arbitrary set of points whose coordinates are given in the
## n_by_2 matrix @var{points}.
##
## @end deftypefn
## Author: Carlo de Falco
## Created: 2012-10-01
function data = bim2c_intrp (msh, n_data, e_data, p)
%% for each point, find the enclosing tetrahedron
[t_list, b_list] = tsearchn (msh.p.', msh.t(1:3, :)', p);
%% only keep points within tetrahedra
invalid = isnan (t_list);
t_list = t_list (! invalid);
ntl = numel (t_list);
b_list = b_list(! invalid, :);
points(invalid,:) = [];
data = [];
if (! isempty (n_data))
data = cat (1, data, squeeze (
sum (reshape (n_data(msh.t(1:3, t_list), :),
[3, ntl, (columns (n_data))]) .*
repmat (b_list.', [1, 1, (columns (n_data))]), 1)));
endif
if (! isempty (e_data))
data = cat (1, data, e_data(t_list, :));
endif
endfunction
%!test
%! msh = bim2c_mesh_properties (msh2m_structured_mesh (linspace (0, 1, 11), linspace (0, 1, 13), 1, 1:4));
%! x = y = linspace (0, 1, 100).';
%! u = msh.p(1, :).';
%! ui = bim2c_intrp (msh, u, [], [x, y]);
%! assert (ui, linspace (0, 1, 100), 10*eps);
bim/inst/bim2c_mesh_properties.m 000644 000765 000000 00000004070 12207456226 017511 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{omesh}]} = @
## bim2c_mesh_properties(@var{imesh})
##
## Compute the properties of @var{imesh} needed by BIM method and append
## them to @var{omesh} as fields.
##
## @seealso{bim2a_reaction, bim2a_advection_diffusion, bim2a_rhs,
## bim2a_laplacian, bim2a_boundary_mass}
## @end deftypefn
function [omesh] = bim2c_mesh_properties(imesh)
## Check input
if nargin != 1
error("bim2c_mesh_properties: wrong number of input parameters.");
elseif !(isstruct(imesh) && isfield(imesh,"p") &&
isfield (imesh,"t") && isfield(imesh,"e"))
error("bim2c_mesh_properties: first input is not a valid mesh structure.");
endif
omesh = imesh;
[omesh.wjacdet,omesh.area,omesh.shg] = ...
msh2m_geometrical_properties(imesh,"wjacdet","area","shg");
endfunction
%!shared mesh
% x = y = linspace(0,1,4);
% [mesh] = msh2m_structured_mesh(x,y,1,1:4);
% [mesh] = bim2c_mesh_properties(mesh);
%!test
% tmp = msh2m_geometrical_properties(mesh,"wjacdet");
% assert(mesh.wjacdet,tmp);
%!test
% tmp = msh2m_geometrical_properties(mesh,"shg");
% assert(mesh.shg,tmp);
%!test
% assert(mesh.area,sum(mesh.wjacdet,1)); bim/inst/bim2c_norm.m 000644 000765 000000 00000010774 12316041364 015256 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006-2013 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Matteo Porro
## -*- texinfo -*-
##
## @deftypefn {Function File} {[@var{norm_u}]} = @
## bim2c_norm(@var{mesh},@var{u},@var{norm_type})
##
## Compute the @var{norm_type}-norm of function @var{u} on the domain described
## by the triangular grid @var{mesh}.
##
## The input function @var{u} can be either a piecewise linear conforming scalar
## function or an elementwise constant scalar or vector function.
##
## The string parameter @var{norm_type} can be one among 'L2', 'H1' and 'inf'.
##
## Should the input function be piecewise constant, the H1 norm will not be
## computed and the function will return an error message.
##
## For the numerical integration of the L2 norm the second order middle point
## quadrature rule is used.
##
## @seealso{bim1c_norm, bim3c_norm}
##
## @end deftypefn
function [norm_u] = bim2c_norm (m, u, norm_type)
## Check input
if (nargin != 3)
error ("bim2c_norm: wrong number of input parameters.");
elseif (! (isstruct (m) && isfield (m,"p")
&& isfield (m, "t")
&& isfield (m, "e")))
error ("bim2c_norm: first input is not a valid mesh structure.");
endif
nnodes = columns (m.p);
nel = columns (m.t);
if (isequal (size (u), [2, nel]))
u = u';
endif
if ((numel (u) != nnodes) && (rows (u) != nel))
error ("bim2c_norm: numel(u) != nnodes and rows(u) != nel.");
endif
if (! (strcmp (norm_type,'L2')
|| strcmp (norm_type,'inf')
|| strcmp (norm_type,'H1')))
error ("bim2c_norm: invalid norm type parameter.");
endif
if (strcmp (norm_type,'inf'))
norm_u = max (abs (u(:)));
else
if (numel (u) == nnodes)
M = __mass_matrix__ (m);
if (strcmp (norm_type, 'H1'))
A = bim2a_laplacian (m, 1, 1);
M += A;
endif
norm_u = sqrt(u' * M * u);
else
if (strcmp (norm_type, 'H1'))
error (["bim2c_norm: cannot compute the H1 norm ", ...
"of an elementwise constant function."]);
endif
norm_u = m.area' * (norm (u, 2, 'rows').^2);
norm_u = sqrt (norm_u);
endif
endif
endfunction
function M = __mass_matrix__ (mesh)
t = mesh.t;
nnodes = columns (mesh.p);
nelem = columns (t);
## Local contributions
Mref = 1/12 * [2 1 1; 1 2 1; 1 1 2];
area = reshape (mesh.area, 1, 1, nelem);
## Computation
for inode = 1:3
for jnode = 1:3
ginode(inode,jnode,:) = t(inode,:);
gjnode(inode,jnode,:) = t(jnode,:);
endfor
endfor
Mloc = area .* Mref;
## assemble global matrix
M = sparse (ginode(:), gjnode(:), Mloc(:), nnodes, nnodes);
endfunction
%!test
%!shared L, V, x, y, m
%! L = rand (1); V = rand (1); x = linspace (0,L,4); y = x;
%! m = msh2m_structured_mesh (x,y,1,1:4);
%! m.area = msh2m_geometrical_properties (m, 'area');
%! m.shg = msh2m_geometrical_properties (m, 'shg');
%! u = V * ones (columns(m.p),1);
%! uinf = bim2c_norm (m, u, 'inf');
%! uL2 = bim2c_norm (m, u, 'L2');
%! uH1 = bim2c_norm (m, u, 'H1');
%! assert ([uinf, uL2, uH1], [V, V*L, V*L], 1e-12);
%!test
%! u = V * (m.p(1,:) + 2*m.p(2,:))';
%! uinf = bim2c_norm (m, u, 'inf');
%! uL2 = bim2c_norm (m, u, 'L2');
%! uH1 = bim2c_norm (m, u, 'H1');
%! assert ([uinf, uL2, uH1],
%! [3*L*V, V*L^2*sqrt(8/3), V*sqrt(8/3*L^4 + 5*L^2)],
%! 1e-12);
%!test
%! u = V * ones (columns(m.t),1);
%! uinf = bim2c_norm (m, u, 'inf');
%! uL2 = bim2c_norm (m, u, 'L2');
%! assert ([uinf, uL2], [V, V*L], 1e-12);
%!test
%! u = V * ones (columns(m.t),1);
%! uvect = [u, 2*u];
%! uinf = bim2c_norm (m, uvect, 'inf');
%! uL2 = bim2c_norm (m, uvect, 'L2');
%! assert ([uinf, uL2], [2*V, V*L*sqrt(5)], 1e-12);
bim/inst/bim2c_pde_gradient.m 000644 000765 000000 00000003432 12316041364 016721 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
##
## @deftypefn {Function File} {[@var{gx},@var{gy}]} = @
## bim2c_pde_gradient(@var{mesh},@var{u})
##
## Compute the gradient of the piecewise linear conforming scalar
## function @var{u}.
##
## @seealso{bim2c_global_flux}
## @end deftypefn
function [gx, gy] = bim2c_pde_gradient(mesh,u)
## Check input
if nargin != 2
error("bim2c_pde_gradient: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim2c_pde_gradient: first input is not a valid mesh structure.");
endif
nnodes = columns(mesh.p);
if (numel (u) != nnodes)
error("bim2c_pde_gradient: length(u) != nnodes.");
endif
shgx = reshape(mesh.shg(1,:,:),3,[]);
gx = sum(shgx.*u(mesh.t(1:3,:)),1);
shgy = reshape(mesh.shg(2,:,:),3,[]);
gy = sum(shgy.*u(mesh.t(1:3,:)),1);
endfunction
bim/inst/bim2c_tri_to_nodes.m 000644 000765 000000 00000005202 12215560060 016756 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2011 Carlo de Falco
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## .
## -*- texinfo -*-
##
## @deftypefn {Function File} {@var{u_nod}} = bim2c_tri_to_nodes (@var{mesh}, @var{u_tri})
## @deftypefnx {Function File} {@var{u_nod}} = bim2c_tri_to_nodes (@var{m_tri}, @var{u_tri})
## @deftypefnx {Function File} {[@var{u_nod}, @var{m_tri}]} = bim2c_tri_to_nodes ( ... )
##
## Compute interpolated values at triangle nodes @var{u_nod} given values at triangle mid-points @var{u_tri}.
## If called with more than one output, also return the interpolation matrix @var{m_tri} such that
## @code{u_nod = m_tri * u_tri}.
## If repeatedly performing interpolation on the same mesh the matrix @var{m_tri} obtained by a previous call
## to @code{bim2c_tri_to_nodes} may be passed as input to avoid unnecessary computations.
##
## @end deftypefn
## Author: Carlo de Falco
## Created: 2011-03-07
function [u_nod, m_tri] = bim2c_tri_to_nodes (m, u_tri)
if (nargout > 1)
if (isstruct (m))
nel = columns (m.t);
nnod = columns (m.p);
ii = m.t(1:3, :);
jj = repmat (1:nel, 3, 1);
vv = repmat (m.area(:)', 3, 1) / 3;
m_tri = bim2a_reaction (m, 1, 1) \ sparse (ii, jj, vv, nnod, nel);
elseif (ismatrix (m))
m_tri = m;
else
error ("bim2c_tri_to_nodes: first input parameter is of incorrect type");
endif
u_nod = m_tri * u_tri;
else
if (isstruct (m))
rhs = bim2a_rhs (m, u_tri, 1);
mass = bim2a_reaction (m, 1, 1);
u_nod = full (mass \ rhs);
elseif (ismatrix (m))
u_nod = m * u_tri;
else
error ("bim2c_tri_to_nodes: first input parameter is of incorrect type");
endif
endif
endfunction
%!test
%! msh = bim2c_mesh_properties (msh2m_structured_mesh (linspace (0, 1, 3), linspace (0, 1, 3), 1, 1:4, "random"));
%! nel = columns (msh.t);
%! nnod = columns (msh.p);
%! u_tri = randn (nel, 1);
%! un1 = bim2c_tri_to_nodes (msh, u_tri);
%! [un2, m] = bim2c_tri_to_nodes (msh, u_tri);
%! assert (un1, un2, 1e-10)
bim/inst/bim2c_unknowns_on_side.m 000644 000765 000000 00000003403 11331503343 017650 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{nodelist}]} = @
## bim2c_unknowns_on_side(@var{mesh},@var{sidelist})
##
## Return the list of the mesh nodes that lie on the geometrical sides
## specified in @var{sidelist}.
##
## @seealso{bim3c_unknown_on_faces, bim2c_pde_gradient,
## bim2c_global_flux}
## @end deftypefn
function [nodelist] = bim2c_unknowns_on_side(mesh, sidelist)
## Check input
if nargin != 2
error("bim2c_unknowns_on_side: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim2c_unknowns_on_side: first input is not a valid mesh structure.");
elseif !isnumeric(sidelist)
error("bim2c_unknowns_on_side: second input is not a valid numeric vector.");
endif
[nodelist] = msh2m_nodes_on_sides(mesh,sidelist);
endfunction
bim/inst/bim3a_advection_diffusion.m 000644 000765 000000 00000007222 12041000520 020276 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2010 Carlo de Falco
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {[@var{A}]} = @
## bim3a_advection_diffusion (@var{mesh}, @var{alpha}, @var{v})
##
## Build the Scharfetter-Gummel stabilized stiffness matrix for a
## diffusion-advection problem.
##
## The equation taken into account is:
##
## - div (@var{alpha} ( grad (u) - grad (@var{v}) u)) = f
##
## where @var{v} is a piecewise linear continuous scalar
## functions and @var{alpha} is a piecewise constant scalar function.
##
## @seealso{bim3a_rhs, bim3a_reaction, bim3a_laplacian, bim3c_mesh_properties}
## @end deftypefn
function SG = bim3a_advection_diffusion (mesh, acoeff, v)
t = mesh.t;
nnodes = columns (mesh.p);
nelem = columns(t);
## Local contributions
Lloc = zeros(4,4,nelem);
epsilonareak = reshape (acoeff .* mesh.area', 1, 1, nelem);
shg = mesh.shg(:,:,:);
## Computation
for inode = 1:4
for jnode = 1:4
ginode(inode,jnode,:) = t(inode,:);
gjnode(inode,jnode,:) = t(jnode,:);
Lloc(inode,jnode,:) = ...
sum( shg(:,inode,:) .* shg(:,jnode,:), 1) .* epsilonareak;
endfor
endfor
vloc = v(t(1:4, :));
[bp12,bm12] = bimu_bernoulli (vloc(2,:)-vloc(1,:));
[bp13,bm13] = bimu_bernoulli (vloc(3,:)-vloc(1,:));
[bp14,bm14] = bimu_bernoulli (vloc(4,:)-vloc(1,:));
[bp23,bm23] = bimu_bernoulli (vloc(3,:)-vloc(2,:));
[bp24,bm24] = bimu_bernoulli (vloc(4,:)-vloc(2,:));
[bp34,bm34] = bimu_bernoulli (vloc(4,:)-vloc(3,:));
bp12 = reshape (bp12, 1, 1, nelem) .* Lloc(1,2,:);
bm12 = reshape (bm12, 1, 1, nelem) .* Lloc(1,2,:);
bp13 = reshape (bp13, 1, 1, nelem) .* Lloc(1,3,:);
bm13 = reshape (bm13, 1, 1, nelem) .* Lloc(1,3,:);
bp14 = reshape (bp14, 1, 1, nelem) .* Lloc(1,4,:);
bm14 = reshape (bm14, 1, 1, nelem) .* Lloc(1,4,:);
bp23 = reshape (bp23, 1, 1, nelem) .* Lloc(2,3,:);
bm23 = reshape (bm23, 1, 1, nelem) .* Lloc(2,3,:);
bp24 = reshape (bp24, 1, 1, nelem) .* Lloc(2,4,:);
bm24 = reshape (bm24, 1, 1, nelem) .* Lloc(2,4,:);
bp34 = reshape (bp34, 1, 1, nelem) .* Lloc(3,4,:);
bm34 = reshape (bm34, 1, 1, nelem) .* Lloc(3,4,:);
## SGloc=[...
## -bm12-bm13-bm14,bp12 ,bp13 ,bp14
## bm12 ,-bp12-bm23-bm24 ,bp23 ,bp24
## bm13 ,bm23 ,-bp13-bp23-bm34,bp34
## bm14 ,bm24 ,bm34 ,-bp14-bp24-bp34...
## ];
Sloc(1,1,:) = -bm12-bm13-bm14;
Sloc(1,2,:) = bp12;
Sloc(1,3,:) = bp13;
Sloc(1,4,:) = bp14;
Sloc(2,1,:) = bm12;
Sloc(2,2,:) = -bp12-bm23-bm24;
Sloc(2,3,:) = bp23;
Sloc(2,4,:) = bp24;
Sloc(3,1,:) = bm13;
Sloc(3,2,:) = bm23;
Sloc(3,3,:) = -bp13-bp23-bm34;
Sloc(3,4,:) = bp34;
Sloc(4,1,:) = bm14;
Sloc(4,2,:) = bm24;
Sloc(4,3,:) = bm34;
Sloc(4,4,:) = -bp14-bp24-bp34;
## assemble global matrix
SG = sparse(ginode(:),gjnode(:),Sloc(:), nnodes, nnodes);
endfunction
bim/inst/bim3a_boundary_mass.m 000644 000765 000000 00000004775 12316041364 017154 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Matteo Porro
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{M}]} = @
## bim3a_boundary_mass(@var{mesh},@var{facelist},@var{nodelist})
##
## Build the lumped boundary mass matrix needed to apply Robin boundary
## conditions.
##
## The vector @var{facelist} contains the list of the faces contributing
## to the mass matrix.
##
## The optional argument @var{nodelist} contains the list of the
## degrees of freedom on the boundary.
##
## @seealso{bim3a_rhs, bim3a_advection_diffusion, bim3a_laplacian,
## bim3a_reaction, bim2a_boundary_mass}
## @end deftypefn
function [M] = bim3a_boundary_mass (mesh, facelist, nodelist)
## Check input
if (nargin > 3)
error ("bim3a_boundary_mass: wrong number of input parameters.");
elseif (! ((isstruct (mesh)) && (isfield (mesh, "p"))
&& (isfield (mesh, "t")) && isfield(mesh, "e")))
error (["bim3a_boundary_mass: first input", ...
" is not a valid mesh structure."]);
elseif (! ((isvector (facelist)) && (isnumeric (facelist))))
error (["bim3a_boundary_mass: second ", ...
"input is not a valid numeric vector."]);
endif
if (nargin < 3)
[nodelist] = bim3c_unknowns_on_faces (mesh, facelist);
endif
p = mesh.p;
t = [];
for ie = facelist
t = [t, mesh.e([1:3 10], mesh.e(10,:) == ie)];
endfor
area = 1/2 * norm (cross (p(:,t(2,:))-p(:,t(1,:)),
p(:,t(3,:))-p(:,t(1,:))),
2, 'columns');
dd = zeros (size (nodelist));
for in = 1:numel (nodelist)
dd (in) = 1/3 * sum (area (any (t(1:3,:) == nodelist(in))));
endfor
M = sparse (diag (dd));
endfunction
bim/inst/bim3a_laplacian.m 000644 000765 000000 00000006707 12316041364 016227 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
## @deftypefn {Function File} @
## {@var{A}} = bim3a_laplacian (@var{mesh}, @var{epsilon}, @var{kappa})
##
## Build the standard finite element stiffness matrix for a diffusion
## problem.
##
## The equation taken into account is:
##
## - (@var{epsilon} * @var{kappa} ( u' ))' = f
##
## where @var{epsilon} is an element-wise constant scalar function,
## while @var{kappa} is a piecewise linear conforming scalar function.
##
## @seealso{bim3a_rhs, bim3a_reaction, bim2a_laplacian, bim3a_laplacian}
## @end deftypefn
function [A] = bim3a_laplacian (mesh,epsilon,kappa)
## Check input
if nargin != 3
error("bim3a_laplacian: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim3a_laplacian: first input is not a valid mesh structure.");
endif
p = mesh.p;
t = mesh.t;
nnodes = columns(p);
nelem = columns(t);
## Turn scalar input to a vector of appropriate size
if isscalar(epsilon)
epsilon = epsilon * ones(nelem,1);
endif
if isscalar(kappa)
kappa = kappa*ones(nnodes,1);
endif
if !( isvector(epsilon) && isvector(kappa) )
error("bim3a_laplacian: coefficients are not valid vectors.");
elseif (numel (epsilon) != nelem)
error("bim3a_laplacian: length of epsilon is not equal to the number of elements.");
elseif (numel (kappa) != nnodes)
error("bim2a_laplacian: length of kappa is not equal to the number of nodes.");
endif
## Local contributions
Lloc = zeros(4,4,nelem);
epsilonareak = reshape (epsilon .* mesh.area',1,1,nelem);
shg = mesh.shg(:,:,:);
## Computation
for inode = 1:4
for jnode = 1:4
ginode(inode,jnode,:) = mesh.t(inode,:);
gjnode(inode,jnode,:) = mesh.t(jnode,:);
Lloc(inode,jnode,:) = sum( kappa(inode) * shg(:,inode,:) .* shg(:,jnode,:),1) .* epsilonareak;
endfor
endfor
## Assembly
A = sparse(ginode(:),gjnode(:),Lloc(:));
endfunction
%!shared mesh,epsilon,kappa,nnodes,nelem
% x = y = z = linspace(0,1,4);
% [mesh] = msh3m_structured_mesh(x,y,z,1,1:6);
% [mesh] = bim3c_mesh_properties(mesh);
% nnodes = columns(mesh.p);
% nelem = columns(mesh.t);
% epsilon = ones(columns(mesh.t),1);
% kappa = ones(columns(mesh.p),1);
%!test
% [A] = bim3a_laplacian(mesh,epsilon,kappa);
% assert(size(A),[nnodes, nnodes]);
%!test
% [A1] = bim3a_laplacian(mesh,3*epsilon,kappa);
% [A2] = bim3a_laplacian(mesh,epsilon,3*kappa);
% assert(A1,A2);
%!test
% [A1] = bim3a_laplacian(mesh,epsilon,kappa);
% [A2] = bim3a_laplacian(mesh,1,1);
% assert(A1,A2);
bim/inst/bim3a_osc_advection_diffusion.m 000644 000765 000000 00000026677 12241245204 021175 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2012 Carlo de Falco
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {[@var{A}]} = bim3a_osc_advection_diffusion (@var{mesh}, @var{alpha}, @var{v})
##
## Build the Scharfetter-Gummel stabilized OSC stiffness
## matrix for a diffusion-advection problem.
##
## For details on the Orthogonal Subdomain Collocation (OSC) method
## see: M.Putti and C.Cordes, SIAM J.SCI.COMPUT. Vol.19(4), pp.1154-1168, 1998.
##
## The equation taken into account is:
##
## - div (@var{alpha} ( grad (u) - grad (@var{v}) u)) = f
##
## where @var{v} is a piecewise linear continuous scalar
## functions and @var{alpha} is a piecewise constant scalar function.
##
## @seealso{bim3a_rhs, bim3a_osc_laplacian, bim3a_reaction, bim3a_laplacian, bim3c_mesh_properties}
## @end deftypefn
function M = bim3a_osc_advection_diffusion (msh, epsilon, v)
## Check input
if (nargin != 3)
print_usage ();
elseif (! (isstruct (msh)
&& isfield (msh, "p")
&& isfield (msh, "t")
&& isfield (msh, "e")))
error (["bim3a_laplacian: first input ", ...
"is not a valid msh structure"]);
endif
nnodes = columns (msh.p);
nelem = columns (msh.t);
## Turn scalar input to a vector of appropriate size
if (isscalar (epsilon))
epsilon = epsilon * ones(nelem, 1);
endif
if (! isvector (epsilon))
error ("bim3a_laplacian: coefficient is not a vector");
elseif (numel (epsilon) != nelem)
error (["bim3a_laplacian: length of epsilon is ", ...
"not equal to the number of mesh elements"]);
endif
## Avoid warnings for broadcasting
warning ("off", "Octave:broadcast", "local")
## Local contributions
Lloc = __osc_local_laplacian__ (msh.p, msh.t, msh.shg, epsilon, msh.area, nnodes, nelem);
## Stabilization
if (isscalar (v))
v = zeros (nelem, 1);
endif
vloc = v(msh.t(1:4, :));
[bp12, bm12] = bimu_bernoulli (vloc(2,:)-vloc(1,:));
[bp13, bm13] = bimu_bernoulli (vloc(3,:)-vloc(1,:));
[bp14, bm14] = bimu_bernoulli (vloc(4,:)-vloc(1,:));
[bp23, bm23] = bimu_bernoulli (vloc(3,:)-vloc(2,:));
[bp24, bm24] = bimu_bernoulli (vloc(4,:)-vloc(2,:));
[bp34, bm34] = bimu_bernoulli (vloc(4,:)-vloc(3,:));
bp12 = reshape (bp12, 1, 1, nelem) .* Lloc(1,2,:);
bm12 = reshape (bm12, 1, 1, nelem) .* Lloc(1,2,:);
bp13 = reshape (bp13, 1, 1, nelem) .* Lloc(1,3,:);
bm13 = reshape (bm13, 1, 1, nelem) .* Lloc(1,3,:);
bp14 = reshape (bp14, 1, 1, nelem) .* Lloc(1,4,:);
bm14 = reshape (bm14, 1, 1, nelem) .* Lloc(1,4,:);
bp23 = reshape (bp23, 1, 1, nelem) .* Lloc(2,3,:);
bm23 = reshape (bm23, 1, 1, nelem) .* Lloc(2,3,:);
bp24 = reshape (bp24, 1, 1, nelem) .* Lloc(2,4,:);
bm24 = reshape (bm24, 1, 1, nelem) .* Lloc(2,4,:);
bp34 = reshape (bp34, 1, 1, nelem) .* Lloc(3,4,:);
bm34 = reshape (bm34, 1, 1, nelem) .* Lloc(3,4,:);
Sloc(1,1,:) = -bm12-bm13-bm14;
Sloc(1,2,:) = bp12;
Sloc(1,3,:) = bp13;
Sloc(1,4,:) = bp14;
Sloc(2,1,:) = bm12;
Sloc(2,2,:) = -bp12-bm23-bm24;
Sloc(2,3,:) = bp23;
Sloc(2,4,:) = bp24;
Sloc(3,1,:) = bm13;
Sloc(3,2,:) = bm23;
Sloc(3,3,:) = -bp13-bp23-bm34;
Sloc(3,4,:) = bp34;
Sloc(4,1,:) = bm14;
Sloc(4,2,:) = bm24;
Sloc(4,3,:) = bm34;
Sloc(4,4,:) = -bp14-bp24-bp34;
## Assembly
for inode = 1:4
for jnode = 1:4
ginode(inode, jnode,:) = msh.t(inode, :);
gjnode(inode, jnode,:) = msh.t(jnode, :);
endfor
endfor
M = sparse (ginode(:), gjnode(:), Sloc(:), nnodes, nnodes);
endfunction
%!shared msh, epsilon, M, nnodes, nelem, x, y, z
%!test
%! msh = bim3c_mesh_properties (msh3m_structured_mesh (0:5, 0:5, 0:5, 1, 1:6));
%! x = msh.p (1, :).';
%! y = msh.p (2, :).';
%! z = msh.p (3, :).';
%! u = ones (size (x));
%! M = bim3a_osc_advection_diffusion (msh, 1, 0);
%! assert (M * u, zeros (size (u)), eps * 100)
%!test
%! u = x;
%! bnd = bim3c_unknowns_on_faces (msh, [1, 2]);
%! int = setdiff (1:columns (msh.p), bnd);
%! assert (M(int, int) * u(int), -M(int, bnd) * u(bnd), 100 * eps)
%!test
%! u = y;
%! bnd = bim3c_unknowns_on_faces (msh, [3, 4]);
%! int = setdiff (1:columns (msh.p), bnd);
%! assert (M(int, int) * u(int), -M(int, bnd) * u(bnd), 100 * eps)
%!test
%! u = z;
%! bnd = bim3c_unknowns_on_faces (msh, [5, 6]);
%! int = setdiff (1:columns (msh.p), bnd);
%! assert (M(int, int) * u(int), -M(int, bnd) * u(bnd), 100 * eps)
%!test
%! u = z;
%! bnd = bim3c_unknowns_on_faces (msh, [5, 6]);
%! int = setdiff (1:columns (msh.p), bnd);
%! M = bim3a_osc_advection_diffusion (msh, pi, 0);
%! assert (M(int, int) * u(int), -M(int, bnd) * u(bnd), 100 * eps)
%!test
%! M = bim3a_osc_advection_diffusion (msh, 1, x);
%! assert (norm (sum (M, 1), inf), 0, eps * 100)
%!test
%! M = bim3a_osc_advection_diffusion (msh, 1, y);
%! assert (norm (sum (M, 1), inf), 0, eps * 100)
%!test
%! M = bim3a_osc_advection_diffusion (msh, 1, z);
%! assert (norm (sum (M, 1), inf), 0, eps * 100)
%!demo
%! gmsh_input = [["Point(1) = {0, 0, 0, .1}; \n"], ...
%! ["Point(2) = {1, 0, 0, .1}; \n"], ...
%! ["Point(3) = {0, -.3, 0, .1}; \n"], ...
%! ["Point(4) = {0, +.3, 0, .1}; \n"], ...
%! ["Point(5) = {1, -.3, 0, .1}; \n"], ...
%! ["Point(6) = {1, 0.3, 0, .1}; \n"], ...
%! ["Point(7) = {0, 0, -.3, .1}; \n"], ...
%! ["Point(8) = {0, 0, +.3, .1}; \n"], ...
%! ["Point(9) = {1, 0, -.3, .1}; \n"], ...
%! ["Point(10) = {1, 0, 0.3, .1}; \n"], ...
%! ["Circle(1) = {4, 1, 7}; \n"], ...
%! ["Circle(2) = {7, 1, 3}; \n"], ...
%! ["Circle(3) = {3, 1, 8}; \n"], ...
%! ["Circle(4) = {8, 1, 4}; \n"], ...
%! ["Circle(5) = {6, 2, 9}; \n"], ...
%! ["Circle(6) = {9, 2, 5}; \n"], ...
%! ["Circle(7) = {5, 2, 10}; \n"], ...
%! ["Circle(8) = {10, 2, 6}; \n"], ...
%! ["Line(9) = {4, 6}; \n"], ...
%! ["Line(10) = {3, 5}; \n"], ...
%! ["Line(11) = {8, 10}; \n"], ...
%! ["Line(12) = {7, 9}; \n"], ...
%! ["Line Loop(13) = {4, 1, 2, 3}; \n"], ...
%! ["Plane Surface(14) = {13}; \n"], ...
%! ["Line Loop(15) = {5, 6, 7, 8}; \n"], ...
%! ["Plane Surface(16) = {15}; \n"], ...
%! ["Line Loop(17) = {9, -8, -11, 4}; \n"], ...
%! ["Ruled Surface(18) = {17}; \n"], ...
%! ["Line Loop(19) = {12, -5, -9, 1}; \n"], ...
%! ["Ruled Surface(20) = {19}; \n"], ...
%! ["Line Loop(21) = {12, 6, -10, -2}; \n"], ...
%! ["Ruled Surface(22) = {21}; \n"], ...
%! ["Line Loop(23) = {11, -7, -10, 3}; \n"], ...
%! ["Ruled Surface(24) = {23}; \n"], ...
%! ["Surface Loop(25) = {18, 20, 22, 16, 24, 14}; \n"], ...
%! ["Volume(26) = {25}; \n"]];
%! fname = tmpnam ();
%! [fid, msg] = fopen (strcat (fname, ".geo"), "w");
%! if (fid < 0); error (msg); endif
%! fputs (fid, gmsh_input);
%! fclose (fid);
%! msh = bim3c_mesh_properties (msh3m_gmsh (fname, "clscale", ".25"));
%! x = msh.p (1, :).';
%! u = x;
%! bnd = bim3c_unknowns_on_faces (msh, [14, 16]);
%! int = setdiff (1:columns (msh.p), bnd);
%! Mosc = bim3a_osc_advection_diffusion (msh, 1, msh.p(1,:)'*0);
%! Mgal = bim3a_advection_diffusion (msh, 1, msh.p(1,:)'*0);
%! u(int) = Mosc(int, int) \ ( - Mosc(int, bnd) * u(bnd));
%! uosc = u;
%! u(int) = Mgal(int, int) \ ( - Mgal(int, bnd) * u(bnd));
%! ugal = u;
%! fname_out = tmpnam ();
%! printf ("saving results to %s \n", strcat (fname_out, ".vtu"));
%! fpl_vtk_raw_write_field (fname_out, msh, {uosc, "u_osc"; ugal, "u_galerkin"}, {});
%! unlink (fname);
%!demo
%! gmsh_input = [["Point(1) = {0, 0, 0, .1}; \n"], ...
%! ["Point(2) = {1, 0, 0, .1}; \n"], ...
%! ["Point(3) = {0, -.3, 0, .1}; \n"], ...
%! ["Point(4) = {0, +.3, 0, .1}; \n"], ...
%! ["Point(5) = {1, -.3, 0, .1}; \n"], ...
%! ["Point(6) = {1, 0.3, 0, .1}; \n"], ...
%! ["Point(7) = {0, 0, -.3, .1}; \n"], ...
%! ["Point(8) = {0, 0, +.3, .1}; \n"], ...
%! ["Point(9) = {1, 0, -.3, .1}; \n"], ...
%! ["Point(10) = {1, 0, 0.3, .1}; \n"], ...
%! ["Circle(1) = {4, 1, 7}; \n"], ...
%! ["Circle(2) = {7, 1, 3}; \n"], ...
%! ["Circle(3) = {3, 1, 8}; \n"], ...
%! ["Circle(4) = {8, 1, 4}; \n"], ...
%! ["Circle(5) = {6, 2, 9}; \n"], ...
%! ["Circle(6) = {9, 2, 5}; \n"], ...
%! ["Circle(7) = {5, 2, 10}; \n"], ...
%! ["Circle(8) = {10, 2, 6}; \n"], ...
%! ["Line(9) = {4, 6}; \n"], ...
%! ["Line(10) = {3, 5}; \n"], ...
%! ["Line(11) = {8, 10}; \n"], ...
%! ["Line(12) = {7, 9}; \n"], ...
%! ["Line Loop(13) = {4, 1, 2, 3}; \n"], ...
%! ["Plane Surface(14) = {13}; \n"], ...
%! ["Line Loop(15) = {5, 6, 7, 8}; \n"], ...
%! ["Plane Surface(16) = {15}; \n"], ...
%! ["Line Loop(17) = {9, -8, -11, 4}; \n"], ...
%! ["Ruled Surface(18) = {17}; \n"], ...
%! ["Line Loop(19) = {12, -5, -9, 1}; \n"], ...
%! ["Ruled Surface(20) = {19}; \n"], ...
%! ["Line Loop(21) = {12, 6, -10, -2}; \n"], ...
%! ["Ruled Surface(22) = {21}; \n"], ...
%! ["Line Loop(23) = {11, -7, -10, 3}; \n"], ...
%! ["Ruled Surface(24) = {23}; \n"], ...
%! ["Surface Loop(25) = {18, 20, 22, 16, 24, 14}; \n"], ...
%! ["Volume(26) = {25}; \n"]];
%! fname = tmpnam ();
%! [fid, msg] = fopen (strcat (fname, ".geo"), "w");
%! if (fid < 0); error (msg); endif
%! fputs (fid, gmsh_input);
%! fclose (fid);
%! msh = bim3c_mesh_properties (msh3m_gmsh (fname, "clscale", ".25"));
%! x = msh.p (1, :).';
%! u = x;
%! bnd = bim3c_unknowns_on_faces (msh, [14, 16]);
%! int = setdiff (1:columns (msh.p), bnd);
%! Mosc = bim3a_osc_advection_diffusion (msh, 1, msh.p(1,:)'*0);
%! Mgal = bim3a_advection_diffusion (msh, 1, msh.p(1,:)'*0);
%! f = bim3a_rhs (msh, 10, 1);
%! u(int) = Mosc(int, int) \ (f(int) - Mosc(int, bnd) * u(bnd));
%! uosc = u;
%! u(int) = Mgal(int, int) \ (f(int) - Mgal(int, bnd) * u(bnd));
%! ugal = u;
%! fname_out = tmpnam ();
%! printf ("saving results to %s \n", strcat (fname_out, ".vtu"));
%! fpl_vtk_raw_write_field (fname_out, msh, {uosc, "u_osc"; ugal, "u_galerkin"}, {});
%! unlink (fname);
bim/inst/bim3a_osc_laplacian.m 000644 000765 000000 00000007310 12241245515 017063 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2012 Carlo de Falco
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## -*- texinfo -*-
## @deftypefn {Function File} @
## {@var{A}} = bim3a_osc_laplacian (@var{mesh}, @var{epsilon})
##
## Build the osc finite element stiffness matrix for a diffusion
## problem.
##
## For details on the Orthogonal Subdomain Collocation (OSC) method
## see: M.Putti and C.Cordes, SIAM J.SCI.COMPUT. Vol.19(4), pp.1154-1168, 1998.
##
## The equation taken into account is:
##
## - div (@var{epsilon} grad (u)) = f
##
## where @var{epsilon} is an element-wise constant scalar function.
##
## @seealso{bim3a_rhs, bim3a_reaction, bim2a_laplacian, bim3a_laplacian}
## @end deftypefn
function M = bim3a_osc_laplacian (msh, epsilon)
## Check input
if (nargin != 2)
print_usage ();
elseif (! (isstruct (msh)
&& isfield (msh, "p")
&& isfield (msh, "t")
&& isfield (msh, "e")))
error (["bim3a_osc_laplacian: first input ", ...
"is not a valid msh structure"]);
endif
nnodes = columns (msh.p);
nelem = columns (msh.t);
## Turn scalar input to a vector of appropriate size
if (isscalar (epsilon))
epsilon = epsilon * ones (nelem, 1);
endif
if (! isvector (epsilon))
error ("bim3a_osc_laplacian: coefficient is not a vector");
elseif (numel (epsilon) != nelem)
error (["bim3a_osc_laplacian: length of epsilon is ", ...
"not equal to the number of mesh elements"]);
endif
## Avoid warnings for broadcasting
warning ("off", "Octave:broadcast", "local")
## Local contributions
Lloc = __osc_local_laplacian__ (msh.p, msh.t, msh.shg,
epsilon, msh.area, nnodes,
nelem);
## Assembly
for inode = 1:4
for jnode = 1:4
ginode(inode, jnode,:) = msh.t(inode, :);
gjnode(inode, jnode,:) = msh.t(jnode, :);
endfor
endfor
M = sparse (ginode(:), gjnode(:), Lloc(:), nnodes, nnodes);
endfunction
%!shared msh, epsilon, M, nnodes, nelem, x, y, z
%!test
%! msh = bim3c_mesh_properties (msh3m_structured_mesh (0:5, 0:5, 0:5, 1, 1:6));
%! x = msh.p (1, :).';
%! y = msh.p (2, :).';
%! z = msh.p (3, :).';
%! u = ones (size (x));
%! M = bim3a_osc_laplacian (msh, 1);
%! assert (M * u, zeros (size (u)), eps * 100)
%!test
%! u = x;
%! bnd = bim3c_unknowns_on_faces (msh, [1, 2]);
%! int = setdiff (1:columns (msh.p), bnd);
%! assert (M(int, int) * u(int), -M(int, bnd) * u(bnd), 100 * eps)
%!test
%! u = y;
%! bnd = bim3c_unknowns_on_faces (msh, [3, 4]);
%! int = setdiff (1:columns (msh.p), bnd);
%! assert (M(int, int) * u(int), -M(int, bnd) * u(bnd), 100 * eps)
%!test
%! u = z;
%! bnd = bim3c_unknowns_on_faces (msh, [5, 6]);
%! int = setdiff (1:columns (msh.p), bnd);
%! assert (M(int, int) * u(int), -M(int, bnd) * u(bnd), 100 * eps)
%!test
%! u = z;
%! bnd = bim3c_unknowns_on_faces (msh, [5, 6]);
%! int = setdiff (1:columns (msh.p), bnd);
%! M = bim3a_osc_laplacian (msh, pi);
%! assert (M(int, int) * u(int), -M(int, bnd) * u(bnd), 100 * eps)
bim/inst/bim3a_reaction.m 000644 000765 000000 00000006240 12316041364 016077 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
## @deftypefn {Function File} @
## {[@var{C}]} = bim3a_reaction (@var{mesh},@var{delta},@var{zeta})
##
## Build the lumped finite element mass matrix for a diffusion
## problem.
##
## The equation taken into account is:
##
## @var{delta} * @var{zeta} * u = f
##
## where @var{delta} is an element-wise constant scalar function, while
## @var{zeta} is a piecewise linear conforming scalar function.
##
## @seealso{bim3a_rhs, bim3a_laplacian, bim2a_reaction, bim3a_reaction}
## @end deftypefn
function [C] = bim3a_reaction (mesh,delta,zeta);
## Check input
if nargin != 3
error("bim3a_reaction: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim3a_reaction: first input is not a valid mesh structure.");
endif
nnodes = columns (mesh.p);
nelem = columns (mesh.t);
## Turn scalar input to a vector of appropriate size
if isscalar(delta)
delta = delta*ones(nelem,1);
endif
if isscalar(zeta)
zeta = zeta*ones(nnodes,1);
endif
if !( isvector(delta) && isvector(zeta) )
error("bim3a_reaction: coefficients are not valid vectors.");
elseif (numel (delta) != nelem)
error("bim3a_: length of alpha is not equal to the number of elements.");
elseif (numel (zeta) != nnodes)
error("bim3a_: length of gamma is not equal to the number of nodes.");
endif
Cloc = zeros(4,nelem);
coeff = zeta(mesh.t(1:4,:));
coeffe = delta;
wjacdet = mesh.wjacdet;
for inode = 1:4
Cloc(inode,:) = coeffe'.*coeff(inode,:).*wjacdet(inode,:);
endfor
gnode = (mesh.t(1:4,:));
## Global matrix
C = sparse(gnode(:),gnode(:),Cloc(:));
endfunction
%!shared mesh,delta,zeta,nnodes,nelem
% x = y = z = linspace(0,1,4);
% [mesh] = msh3m_structured_mesh(x,y,z,1,1:6);
% [mesh] = bim3c_mesh_properties(mesh);
% nnodes = columns(mesh.p);
% nelem = columns(mesh.t);
% delta = ones(columns(mesh.t),1);
% zeta = ones(columns(mesh.p),1);
%!test
% [C] = bim3a_reaction(mesh,delta,zeta);
% assert(size(C),[nnodes, nnodes]);
%!test
% [C1] = bim3a_reaction(mesh,3*delta,zeta);
% [C2] = bim3a_reaction(mesh,delta,3*zeta);
% assert(C1,C2);
%!test
% [C1] = bim2a_reaction(mesh,3*delta,zeta);
% [C2] = bim2a_reaction(mesh,3,1);
% assert(C1,C2);
bim/inst/bim3a_rhs.m 000644 000765 000000 00000006040 12316041364 015065 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{b}]} = @
## bim3a_rhs (@var{mesh}, @var{f}, @var{g})
##
## Build the finite element right-hand side of a diffusion problem
## employing mass-lumping.
##
## The equation taken into account is:
##
## @var{delta} * u = @var{f} * @var{g}
##
## where @var{f} is an element-wise constant scalar function, while
## @var{g} is a piecewise linear conforming scalar function.
##
## @seealso{bim3a_reaction, bim3_laplacian, bim1a_reaction,
## bim2a_reaction}
## @end deftypefn
function [b] = bim3a_rhs (mesh,f,g);
## Check input
if nargin != 3
error("bim3a_rhs: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim3a_rhs: first input is not a valid mesh structure.");
endif
nnodes = columns (mesh.p);
nelem = columns (mesh.t);
## Turn scalar input to a vector of appropriate size
if isscalar(f)
f = f*ones(nelem,1);
endif
if isscalar(g)
g = g*ones(nnodes,1);
endif
if !( isvector(f) && isvector(g) )
error("bim3a_rhs: coefficients are not valid vectors.");
elseif (numel (f) != nelem)
error("bim3a_rhs: length of f is not equal to the number of elements.");
elseif (numel (g) != nnodes)
error("bim3a_rhs: length of g is not equal to the number of nodes.");
endif
bloc = zeros(4,nelem);
coeff = g(mesh.t(1:4,:));
coeffe = f;
wjacdet = mesh.wjacdet;
for inode = 1:4
bloc(inode,:) = coeffe'.*coeff(inode,:).*wjacdet(inode,:);
endfor
gnode = (mesh.t(1:4,:));
## Global matrix
b = sparse(gnode(:),1,bloc(:));
endfunction
%!shared mesh,f,g,nnodes,nelem
% x = y = z = linspace(0,1,4);
% [mesh] = msh3m_structured_mesh(x,y,z,1,1:6);
% [mesh] = bim3c_mesh_properties(mesh);
% nnodes = columns(mesh.p);
% nelem = columns(mesh.t);
% g = ones(columns(mesh.t),1);
% f = ones(columns(mesh.p),1);
%!test
% [b] = bim3a_rhs(mesh,f,g);
% assert(size(b),[nnodes, 1]);
%!test
% [b1] = bim3a_rhs(mesh,3*f,g);
% [b2] = bim3a_rhs(mesh,f,3*g);
% assert(b1,b2);
%!test
% [b1] = bim2a_rhs(mesh,3*f,g);
% [b2] = bim2a_rhs(mesh,3,1);
% assert(b1,b2);
bim/inst/bim3c_global_flux.m 000644 000765 000000 00000010137 12215560060 016570 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2012 Carlo de Falco
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {[@var{F}]} = @
## bim3c_global_flux (@var{mesh}, @var{u}, @var{alpha}, @var{v})
##
## Compute the flux associated with the Scharfetter-Gummel approximation
## of the scalar field @var{u}.
##
## The vector field is defined as:
##
## F =- @var{alpha} ( grad (u) - grad (@var{v}) u )
##
## where @var{v} is a piecewise linear continuous scalar
## functions and @var{alpha} is a piecewise constant scalar function.
##
## @seealso{bim3a_rhs, bim3a_reaction, bim3a_laplacian, bim3c_mesh_properties}
## @end deftypefn
function F = bim3c_global_flux (mesh, u, acoeff, v)
t = mesh.t;
nelem = columns (mesh.t);
F = zeros (3, nelem);
## Local contributions
Lloc = zeros (4,4,nelem);
epsilonareak = reshape (acoeff .* mesh.area', 1, 1, nelem);
shg = mesh.shg(:,:,:);
## Computation
for inode = 1:4
for jnode = 1:4
ginode(inode,jnode,:) = t(inode,:);
gjnode(inode,jnode,:) = t(jnode,:);
Lloc(inode,jnode,:) = sum (shg(:,inode,:) .* shg(:,jnode,:), 1) ...
.* epsilonareak;
endfor
endfor
uloc = u(t(1:4, :));
vloc = v(t(1:4, :));
[bp12,bm12] = bimu_bernoulli (vloc(2,:)-vloc(1,:));
[bp13,bm13] = bimu_bernoulli (vloc(3,:)-vloc(1,:));
[bp14,bm14] = bimu_bernoulli (vloc(4,:)-vloc(1,:));
[bp23,bm23] = bimu_bernoulli (vloc(3,:)-vloc(2,:));
[bp24,bm24] = bimu_bernoulli (vloc(4,:)-vloc(2,:));
[bp34,bm34] = bimu_bernoulli (vloc(4,:)-vloc(3,:));
bp12 = reshape (bp12, 1, 1, nelem) .* Lloc(1,2,:);
bm12 = reshape (bm12, 1, 1, nelem) .* Lloc(1,2,:);
bp13 = reshape (bp13, 1, 1, nelem) .* Lloc(1,3,:);
bm13 = reshape (bm13, 1, 1, nelem) .* Lloc(1,3,:);
bp14 = reshape (bp14, 1, 1, nelem) .* Lloc(1,4,:);
bm14 = reshape (bm14, 1, 1, nelem) .* Lloc(1,4,:);
bp23 = reshape (bp23, 1, 1, nelem) .* Lloc(2,3,:);
bm23 = reshape (bm23, 1, 1, nelem) .* Lloc(2,3,:);
bp24 = reshape (bp24, 1, 1, nelem) .* Lloc(2,4,:);
bm24 = reshape (bm24, 1, 1, nelem) .* Lloc(2,4,:);
bp34 = reshape (bp34, 1, 1, nelem) .* Lloc(3,4,:);
bm34 = reshape (bm34, 1, 1, nelem) .* Lloc(3,4,:);
## SGloc=[...
## -bm12-bm13-bm14,bp12 ,bp13 ,bp14
## bm12 ,-bp12-bm23-bm24 ,bp23 ,bp24
## bm13 ,bm23 ,-bp13-bp23-bm34,bp34
## bm14 ,bm24 ,bm34 ,-bp14-bp24-bp34
## ];
Sloc(1,1,:) = -bm12-bm13-bm14;
Sloc(1,2,:) = bp12;
Sloc(1,3,:) = bp13;
Sloc(1,4,:) = bp14;
Sloc(2,1,:) = bm12;
Sloc(2,2,:) = -bp12-bm23-bm24;
Sloc(2,3,:) = bp23;
Sloc(2,4,:) = bp24;
Sloc(3,1,:) = bm13;
Sloc(3,2,:) = bm23;
Sloc(3,3,:) = -bp13-bp23-bm34;
Sloc(3,4,:) = bp34;
Sloc(4,1,:) = bm14;
Sloc(4,2,:) = bm24;
Sloc(4,3,:) = bm34;
Sloc(4,4,:) = -bp14-bp24-bp34;
r = zeros (4, nelem);
f = zeros (3, nelem);
for iel = 1:nelem
r(:,iel) = Sloc(:,:,iel) * uloc(:,iel);
f(:,iel) = Lloc(1:3, 1:3, iel) \ r(1:3, iel);
F(:,iel) = shg(:,1:3, iel) * f(:, iel);
endfor
endfunction
%!test
%! N = 10; pp = linspace (0, 1, N); msh = bim3c_mesh_properties (msh3m_structured_mesh (pp, pp, pp, 1, 1:6));
%! u = ones (N^3, 1);
%! v = ones (N^3, 1);
%! alpha = ones (columns (msh.t), 1);
%! F = bim3c_global_flux (msh, u, alpha, v);
%! assert (norm (F(:), inf), 0, 100*eps);
bim/inst/bim3c_intrp.m 000644 000765 000000 00000004227 12042340077 015433 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2011, 2012 Carlo de Falco
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## .
## -*- texinfo -*-
##
## @deftypefn {Function File} {@var{data}} = bim3c_intrp (@var{msh}, @var{n_data}, @var{e_data}, @var{points})
##
## Compute interpolated values of node centered multicomponent node centered field @var{n_data} and
## cell centered field @var{n_data} at an arbitrary set of points whos coordinates are given in the
## n_by_3 matrix @var{points}.
##
## @end deftypefn
## Author: Carlo de Falco
## Created: 2012-10-01
function data = bim3c_intrp (msh, n_data, e_data, p)
%% for each point, find the enclosing tetrahedron
[t_list, b_list] = tsearchn (msh.p.', msh.t(1:4, :)', p);
%% only keep points within tetrahedra
invalid = isnan (t_list);
t_list = t_list (! invalid);
ntl = numel (t_list);
b_list = b_list(! invalid, :);
points(invalid,:) = [];
data = [];
if (! isempty (n_data))
data = cat (1, data, squeeze (
sum (reshape (n_data(msh.t(1:4, t_list), :),
[4, ntl, (columns (n_data))]) .*
repmat (b_list.', [1, 1, (columns (n_data))]), 1)));
endif
if (! isempty (e_data))
data = cat (1, data, e_data(t_list, :));
endif
endfunction
%!test
%! msh = bim3c_mesh_properties (msh3m_structured_mesh (linspace (0, 1, 11), linspace (0, 1, 9), linspace (0, 1, 13), 1, 1:6));
%! x = y = z = linspace (0, 1, 100).';
%! u = msh.p(1, :).';
%! ui = bim3c_intrp (msh, u, [], [x, y, z]);
%! assert (ui, linspace (0, 1, 100), 10*eps); bim/inst/bim3c_mesh_properties.m 000644 000765 000000 00000004157 12020717630 017510 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{omesh}]} = @
## bim3c_mesh_properties(@var{imesh})
##
## Compute the properties of @var{imesh} needed by BIM method and append
## them to @var{omesh} as fields.
##
## @seealso{bim3a_reaction, bim3a_rhs, bim3a_laplacian}
## @end deftypefn
function [omesh] = bim3c_mesh_properties (imesh)
## Check input
if nargin != 1
error("bim3c_mesh_properties: wrong number of input parameters.");
elseif (! isstruct (imesh) || any (! isfield (imesh, {"p", "e", "t"})))
error ("bim3c_mesh_properties: first input is not a valid mesh structure.");
endif
## Compute properties
omesh = imesh;
[omesh.wjacdet,omesh.area,omesh.shg,omesh.shp] = ...
msh3m_geometrical_properties (imesh, "wjacdet", "area", "shg", "shp");
endfunction
%!shared mesh
% x = y = z = linspace(0,1,4);
% mesh = msh3m_structured_mesh(x,y,z,1,1:6);
% mesh = bim3c_mesh_properties (mesh);
%!test
% tmp = msh3m_geometrical_properties (mesh, "wjacdet");
% assert(mesh.wjacdet,tmp);
%!test
% tmp = msh3m_geometrical_properties(mesh,"shg");
% assert(mesh.shg,tmp);
%!test
% tmp = msh3m_geometrical_properties(mesh,"shp");
% assert(mesh.shp,tmp);
%!test
% assert(mesh.area,sum(mesh.wjacdet,1));
bim/inst/bim3c_norm.m 000644 000765 000000 00000011552 12316041364 015252 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006-2013 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Matteo Porro
## -*- texinfo -*-
##
## @deftypefn {Function File} {[@var{norm_u}]} = @
## bim3c_norm(@var{mesh},@var{u},@var{norm_type})
##
## Compute the @var{norm_type}-norm of function @var{u} on the domain described
## by the tetrahedral grid @var{mesh}.
##
## The input function @var{u} can be either a piecewise linear conforming scalar
## function or an elementwise constant scalar or vector function.
##
## The string parameter @var{norm_type} can be one among 'L2', 'H1' and 'inf'.
##
## Should the input function be piecewise constant, the H1 norm will not be
## computed and the function will return an error message.
##
## For the numerical integration of the L2 norm the second order quadrature rule
## by Keast is used (ref. P. Keast, Moderate degree tetrahedral quadrature
## formulas, CMAME 55: 339-348 1986).
##
## @seealso{bim1c_norm, bim2c_norm}
##
## @end deftypefn
function [norm_u] = bim3c_norm (m, u, norm_type)
## Check input
if (nargin != 3)
error ("bim3c_norm: wrong number of input parameters.");
elseif (! (isstruct (m) && isfield (m,"p"))
&& isfield (m, "t") && isfield (m, "e"))
error ("bim3c_norm: first input is not a valid mesh structure.");
endif
nnodes = columns (m.p);
nel = columns (m.t);
if (isequal (size (u), [3, nel]))
u = u';
endif
if ((numel (u) != nnodes) && (rows (u) != nel))
error ("bim3c_norm: length(u) != nnodes and rows(u) != nel.");
endif
if (! (strcmp (norm_type,'L2')
|| strcmp (norm_type,'inf')
|| strcmp (norm_type,'H1')))
error ("bim3c_norm: invalid norm type parameter.");
endif
if (strcmp (norm_type,'inf'))
norm_u = max (abs (u(:)));
else
if (numel (u) == nnodes)
M = __mass_matrix__ (m);
if (strcmp (norm_type, 'H1'))
A = bim3a_laplacian (m, 1, 1);
M += A;
endif
norm_u = sqrt(u' * M * u);
else
if (strcmp (norm_type, 'H1'))
error (["bim3c_norm: cannot compute the H1 norm", ...
"of an elementwise constant function."]);
endif
norm_u = m.area * (norm (u', 2, 'cols').^2)';
norm_u = sqrt (norm_u);
endif
endif
endfunction
function M = __mass_matrix__ (mesh)
t = mesh.t;
nnodes = columns (mesh.p);
nelem = columns (t);
## Local contributions
a = (5 + 3 * sqrt (5)) / 20; b = (5 - sqrt (5)) / 20;
l1 = (1 - 3*b)^2 + 3*(1 - 2*b - a)^2;
l2 = (1 - 3*b)*b + (1 - 2*b - a)*(a + 2*b);
Mref = 1/4 * [l1 l2 l2 l2;
l2 l1 l2 l2;
l2 l2 l1 l2;
l2 l2 l2 l1];
area = reshape (mesh.area, 1, 1, nelem);
## Computation
for inode = 1:4
for jnode = 1:4
ginode(inode,jnode,:) = t(inode,:);
gjnode(inode,jnode,:) = t(jnode,:);
endfor
endfor
Mloc = area .* Mref;
## assemble global matrix
M = sparse (ginode(:), gjnode(:), Mloc(:), nnodes, nnodes);
endfunction
%!test
%!shared L, V, x, y, z, m
%! L = rand (1); V = rand (1); x = linspace (0,L,4); y = x; z = x;
%! m = msh3m_structured_mesh (x,y,z,1,1:6);
%! m.area = msh3m_geometrical_properties (m, 'area');
%! m.shg = msh3m_geometrical_properties (m, 'shg');
%! u = V * ones (columns(m.p),1);
%! uinf = bim3c_norm (m, u, 'inf');
%! uL2 = bim3c_norm (m, u, 'L2');
%! uH1 = bim3c_norm (m, u, 'H1');
%! assert ([uinf, uL2, uH1], [V, V*sqrt(L^3), V*sqrt(L^3)], 1e-12);
%!test
%! u = V * (m.p(1,:) + 2*m.p(2,:) + 3*m.p(3,:))';
%! uinf = bim3c_norm (m, u, 'inf');
%! uL2 = bim3c_norm (m, u, 'L2');
%! uH1 = bim3c_norm (m, u, 'H1');
%! assert ([uinf, uL2, uH1],
%! [6*L*V, V*sqrt(61/6*L^5), V*sqrt(61/6*L^5 + 14*L^3)],
%! 1e-12);
%!test
%! u = V * ones (columns(m.t),1);
%! uinf = bim3c_norm (m, u, 'inf');
%! uL2 = bim3c_norm (m, u, 'L2');
%! assert ([uinf, uL2], [V, V*sqrt(L^3)], 1e-12);
%!test
%! u = V * ones (columns(m.t),1);
%! uvect = [u, 2*u, 3*u];
%! uinf = bim3c_norm (m, uvect, 'inf');
%! uL2 = bim3c_norm (m, uvect, 'L2');
%! assert ([uinf, uL2], [3*V, V*sqrt(14*L^3)], 1e-12);
bim/inst/bim3c_pde_gradient.m 000644 000765 000000 00000003526 12316042400 016716 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
##
## @deftypefn {Function File} {[@var{gx}, @var{gy}, @var{gz}]} = @
## bim3c_pde_gradient(@var{mesh},@var{u})
##
## Compute the gradient of the piecewise linear conforming scalar
## function @var{u}.
##
## @seealso{bim3c_global_flux}
## @end deftypefn
function [gx, gy, gz] = bim3c_pde_gradient (mesh, u)
## Check input
if (nargin != 2)
error("bim3c_pde_gradient: wrong number of input parameters.");
elseif (! (isstruct (mesh) && isfield (mesh,"p")) &&
isfield (mesh, "t") && isfield(mesh, "e"))
error ("bim3c_pde_gradient: first input is not a valid mesh structure.");
endif
nnodes = columns (mesh.p);
if (numel (u) != nnodes)
error ("bim3c_pde_gradient: length(u) != nnodes.");
endif
gx = sum (squeeze (mesh.shg(1,:,:)) .* u(mesh.t(1:4,:)), 1);
gy = sum (squeeze (mesh.shg(2,:,:)) .* u(mesh.t(1:4,:)), 1);
gz = sum (squeeze (mesh.shg(3,:,:)) .* u(mesh.t(1:4,:)), 1);
endfunction
bim/inst/bim3c_tri_to_nodes.m 000644 000765 000000 00000005226 12215560060 016765 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2011, 2012 Carlo de Falco
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## .
## -*- texinfo -*-
##
## @deftypefn {Function File} {@var{u_nod}} = bim3c_tri_to_nodes (@var{mesh}, @var{u_tri})
## @deftypefnx {Function File} {@var{u_nod}} = bim3c_tri_to_nodes (@var{m_tri}, @var{u_tri})
## @deftypefnx {Function File} {[@var{u_nod}, @var{m_tri}]} = bim3c_tri_to_nodes ( ... )
##
## Compute interpolated values at triangle nodes @var{u_nod} given values at tetrahedral centers of mass @var{u_tri}.
## If called with more than one output, also return the interpolation matrix @var{m_tri} such that
## @code{u_nod = m_tri * u_tri}.
## If repeatedly performing interpolation on the same mesh the matrix @var{m_tri} obtained by a previous call
## to @code{bim2c_tri_to_nodes} may be passed as input to avoid unnecessary computations.
##
## @end deftypefn
## Author: Carlo de Falco
## Created: 2011-03-07
function [u_nod, m_tri] = bim3c_tri_to_nodes (m, u_tri)
if (nargout > 1)
if (isstruct (m))
nel = columns (m.t);
nnod = columns (m.p);
ii = m.t(1:4, :);
jj = repmat (1:nel, 4, 1);
vv = repmat (m.area(:)', 4, 1) / 4;
m_tri = bim3a_reaction (m, 1, 1) \ sparse (ii, jj, vv, nnod, nel);
elseif (ismatrix (m))
m_tri = m;
else
error ("bim3c_tri_to_nodes: first input parameter is of incorrect type");
endif
u_nod = m_tri * u_tri;
else
if (isstruct (m))
rhs = bim3a_rhs (m, u_tri, 1);
mass = bim3a_reaction (m, 1, 1);
u_nod = full (mass \ rhs);
elseif (ismatrix (m))
u_nod = m * u_tri;
else
error ("bim3c_tri_to_nodes: first input parameter is of incorrect type");
endif
endif
endfunction
%!test
%! msh = bim3c_mesh_properties (msh3m_structured_mesh (linspace (0, 1, 31), linspace (0, 1, 13), linspace (0, 1, 13), 1, 1:6));
%! nel = columns (msh.t);
%! nnod = columns (msh.p);
%! u_tri = randn (nel, 1);
%! un1 = bim3c_tri_to_nodes (msh, u_tri);
%! [un2, m] = bim3c_tri_to_nodes (msh, u_tri);
%! assert (un1, un2, 1e-10)
bim/inst/bim3c_unknowns_on_faces.m 000644 000765 000000 00000004035 11331503343 020010 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{nodelist}]} = @
## bim3c_unknowns_on_faces(@var{mesh},@var{facelist})
##
## Return the list of the mesh nodes that lie on the geometrical faces
## specified in @var{facelist}.
##
## @seealso{bim3c_unknown_on_faces, bim2c_pde_gradient,
## bim2c_global_flux}
##
## @end deftypefn
function [nodelist] = bim3c_unknowns_on_faces(mesh,facelist)
## Check input
if nargin != 2
error("bim3c_unknowns_on_faces: wrong number of input parameters.");
elseif !(isstruct(mesh) && isfield(mesh,"p") &&
isfield (mesh,"t") && isfield(mesh,"e"))
error("bim3c_unknowns_on_faces: first input is not a valid mesh structure.");
elseif !isnumeric(facelist)
error("bim3c_unknowns_on_faces: second input is not a valid numeric vector.");
endif
[nodelist] = msh3m_nodes_on_faces(mesh,facelist);
endfunction
%!shared mesh
% x = y = z = linspace(0,1,2);
% [mesh] = msh3m_structured_mesh(x,y,z,1,1:6);
%!test
% assert( bim3c_unknowns_on_faces(mesh, 1),[1 2 5 6] )
%!test
% assert( bim3c_unknowns_on_faces(mesh, 2),[3 4 7 8] )
%!test
% assert( bim3c_unknowns_on_faces(mesh, [1 2]),1:8) bim/inst/bimu_bernoulli.m 000644 000765 000000 00000004137 11365772115 016242 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {[@var{bp}, @var{bn}]} = bimu_bernoulli (@var{x})
##
## Compute the values of the Bernoulli function corresponding to @var{x}
## and - @var{x} arguments.
##
## @seealso{bimu_logm}
## @end deftypefn
function [bp,bn] = bimu_bernoulli(x)
## Check input
if nargin != 1
error("bimu_bernoulli: wrong number of input parameters.");
endif
xlim= 1e-2;
ax = abs(x);
bp = zeros(size(x));
bn = bp;
block1 = find(~ax);
block21 = find((ax>80)&x>0);
block22 = find((ax>80)&x<0);
block3 = find((ax<=80)&(ax>xlim));
block4 = find((ax<=xlim)&(ax~=0));
## X=0
bp(block1)=1.;
bn(block1)=1.;
## ASYMPTOTICS
bp(block21)=0.;
bn(block21)=x(block21);
bp(block22)=-x(block22);
bn(block22)=0.;
## INTERMEDIATE VALUES
bp(block3)=x(block3)./(exp(x(block3))-1);
bn(block3)=x(block3)+bp(block3);
## SMALL VALUES
if(any(block4))jj=1;
fp=1.*ones(size(block4));
fn=fp;
df=fp;
segno=1.;
while (norm(df,inf) > eps),
jj=jj+1;
segno=-segno;
df=df.*x(block4)/jj;
fp=fp+df;
fn=fn+segno*df;
endwhile;
bp(block4)=1./fp;
bn(block4)=1./fn;
endif
endfunction
bim/inst/bimu_logm.m 000644 000765 000000 00000003121 11331503343 015161 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2006,2007,2008,2009,2010 Carlo de Falco, Massimiliano Culpo
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## author: Massimiliano Culpo
## -*- texinfo -*-
## @deftypefn {Function File} @
## {[@var{T}]} = bimu_logm (@var{t1},@var{t2})
##
## Input:
## @itemize @minus
## @item @var{t1}:
## @item @var{t2}:
## @end itemize
##
## Output:
## @itemize @minus
## @item @var{T}:
## @end itemize
##
## @seealso{bimu_bern}
## @end deftypefn
function [T] = bimu_logm(t1,t2)
## Check input
if nargin != 2
error("bimu_logm: wrong number of input parameters.");
elseif size(t1) != size(t2)
error("bimu_logm: t1 and t2 are of different size.");
endif
T = zeros(size(t2));
sing = abs(t2-t1)< 100*eps ;
T(sing) = (t2(sing)+t1(sing))/2;
T(~sing) = (t2(~sing)-t1(~sing))./log(t2(~sing)./t1(~sing));
endfunction
bim/inst/private/ 000755 000765 000000 00000000000 12420212272 014504 5 ustar 00carlo wheel 000000 000000 bim/inst/private/__osc_local_laplacian__.m 000644 000765 000000 00000010045 12254261047 021431 0 ustar 00carlo wheel 000000 000000 ## Copyright (C) 2012 Carlo de Falco
##
## This file is part of:
## BIM - Diffusion Advection Reaction PDE Solver
##
## BIM is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## BIM is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with BIM; If not, see .
##
## author: Carlo de Falco
## -*- texinfo -*-
##
## @deftypefn {Function File} @
## {@var{Lloc}} = __osc_local_laplacian__ @
## (@var{p}, @var{t}, @var{shg}, @var{epsilon}, @var{area}, @var{nnodes}, @var{nelem})
##
## Unocumented private function.
##
## @end deftypefn
function Lloc = __osc_local_laplacian__ (p, t, shg, epsilon, area, nnodes, nelem)
Lloc = zeros (4, 4, nelem);
epsilonbyareak = epsilon(:) ./ abs (area(:)) / 48;
A = zeros (3, 4, nelem);
## Computation
for inode = 1:4
A(:, inode, :) = 3 * abs (area) .* squeeze (shg (:, inode, :));
endfor
Ann = squeeze (sum (A .^ 2, 1));
r12 = p(:, t (2, :)) - p(:, t (1, :));
r13 = p(:, t (3, :)) - p(:, t (1, :));
r14 = p(:, t (4, :)) - p(:, t (1, :));
r23 = p(:, t (3, :)) - p(:, t (2, :));
r24 = p(:, t (4, :)) - p(:, t (2, :));
r34 = p(:, t (4, :)) - p(:, t (3, :));
s12 = - epsilonbyareak .* (2 * (dot (r13, r23, 1) .*
dot (r14, r24, 1))(:) +
squeeze (dot (A(:, 3, :), A(:, 4, :), 1)) .*
(dot ( r13, r23, 1) .^ 2 ./ Ann(4, :) +
dot ( r14, r24, 1).^ 2 ./ Ann(3, :))(:));
s13 = - epsilonbyareak .* (2 * (dot (r12, -r23, 1) .*
dot (r14, r34, 1))(:) +
squeeze (dot (A(:, 2, :), A(:, 4, :), 1)) .*
(dot ( r12, -r23, 1) .^ 2 ./ Ann(4, :) +
dot ( r14, r34, 1).^ 2 ./ Ann(2, :))(:));
s14 = - epsilonbyareak .* (2 * (dot ( r12, -r24, 1) .*
dot ( r13, -r34, 1))(:) +
squeeze (dot (A(:, 2, :), A(:, 3, :), 1)) .*
(dot ( r12, -r24, 1) .^ 2 ./ Ann(3, :) +
dot ( r13, -r34, 1).^ 2 ./ Ann(2, :))(:));
s23 = - epsilonbyareak .* (2 * (dot (-r12, -r13, 1) .*
dot ( r24, r34, 1))(:) +
squeeze (dot (A(:, 1, :), A(:, 4, :), 1)) .*
(dot (-r12, -r13, 1) .^ 2 ./ Ann(4, :) +
dot ( r24, r34, 1).^ 2 ./ Ann(1, :))(:));
s24 = - epsilonbyareak .* (2 * (dot (-r12, -r14, 1) .*
dot ( r23, -r34, 1))(:) +
squeeze (dot (A(:, 1, :), A(:, 3, :), 1)) .*
(dot (-r12, -r14, 1) .^ 2 ./ Ann(3, :) +
dot ( r23, -r34, 1).^ 2 ./ Ann(1, :))(:));
s34 = - epsilonbyareak .* (2 * (dot (-r13, -r14, 1) .*
dot (-r23, -r24, 1))(:) +
squeeze (dot (A(:, 1, :), A(:, 2, :), 1)) .*
(dot (-r13, -r14, 1) .^ 2 ./ Ann(2, :) +
dot (-r23, -r24, 1).^ 2 ./ Ann(1, :))(:));
Lloc(1, 2, :) = s12; Lloc(2, 1, :) = s12;
Lloc(1, 3, :) = s13; Lloc(3, 1, :) = s13;
Lloc(1, 4, :) = s14; Lloc(4, 1, :) = s14;
Lloc(1, 1, :) = -(s12+s13+s14);
Lloc(2, 3, :) = s23; Lloc(3, 2, :) = s23;
Lloc(2, 4, :) = s24; Lloc(4, 2, :) = s24;
Lloc(2, 2, :) = -(s12+s23+s24);
Lloc(3, 4, :) = s34; Lloc(4, 3, :) = s34;
Lloc(3, 3, :) = -(s13+s23+s34);
Lloc(4, 4, :) = -(s14+s24+s34);
endfunction
bim/doc/fiume.geo 000644 000765 000000 00000001106 12020717630 014425 0 ustar 00carlo wheel 000000 000000 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};
bim/doc/tutorial.html 000644 000765 000000 00000006036 11353166740 015374 0 ustar 00carlo wheel 000000 000000
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.
Create the mesh and precompute the mesh properties
The geometry of the domain was created using gmsh and is stored in the file fiume.geo
[mesh] = msh2m_gmsh("fiume","scale",1,"clscale",.1);
[mesh] = bim2c_mesh_properties(mesh);
Construct an initial guess
For a linear problem only the values at boundary nodes are actually relevant
xu = mesh.p(1,:).';
yu = mesh.p(2,:).';
nelems = columns(mesh.t);
nnodes = columns(mesh.p);
uin = 3*xu;
Set the coefficients for the problem:
-div ( \alpha \gamma ( \eta \nabla u - \beta u ) )+ \delta \zeta u = f g
epsilon = .1;
alfa = ones(nelems,1);
gamma = ones(nnodes,1);
eta = epsilon*ones(nnodes,1);
beta = xu+yu;
delta = ones(nelems,1);
zeta = ones(nnodes,1);
f = ones(nelems,1);
g = ones(nnodes,1);
Construct the discretized operators
AdvDiff = bim2a_advection_diffusion(mesh,alfa,gamma,eta,beta);
Mass = bim2a_reaction(mesh,delta,zeta);
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
Dlist = bim2c_unknowns_on_side(mesh, [8 18]); ## DIRICHLET NODES LIST
Nlist = bim2c_unknowns_on_side(mesh, [23 24]); ## NEUMANN NODES LIST
Nlist = setdiff(Nlist,Dlist);
Fn = zeros(length(Nlist),1); ## PRESCRIBED NEUMANN FLUXES
Ilist = setdiff(1:length(uin),union(Dlist,Nlist)); ## INTERNAL NODES LIST
Add = A(Dlist,Dlist);
Adn = A(Dlist,Nlist); ## shoud be all zeros hopefully!!
Adi = A(Dlist,Ilist);
And = A(Nlist,Dlist); ## shoud be all zeros hopefully!!
Ann = A(Nlist,Nlist);
Ani = A(Nlist,Ilist);
Aid = A(Ilist,Dlist);
Ain = A(Ilist,Nlist);
Aii = A(Ilist,Ilist);
bd = b(Dlist);
bn = b(Nlist);
bi = b(Ilist);
ud = uin(Dlist);
un = uin(Nlist);
ui = uin(Ilist);
Solve for the displacements
temp = [Ann Ani ; Ain Aii ] \ [ Fn+bn-And*ud ; bi-Aid*ud];
un = temp(1:length(un));
ui = temp(length(un)+1:end);
u(Dlist) = ud;
u(Ilist) = ui;
u(Nlist) = un;
Compute the fluxes through Dirichlet sides
Fd = Add * ud + Adi * ui + Adn*un - 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,alfa,gamma,eta,beta);
Save data for later visualization
fpl_dx_write_field("dxdata",mesh,[gx; gy]',"Gradient",1,2,1);
fpl_vtk_write_field ("vtkdata", mesh, {}, {[gx; gy]', "Gradient"}, 1);