pax_global_header00006660000000000000000000000064142664176460014532gustar00rootroot0000000000000052 comment=022f699bc2b27be03dc74fa8d09432f528e5c765 bim-1.1.6/000077500000000000000000000000001426641764600123065ustar00rootroot00000000000000bim-1.1.6/.hgignore000066400000000000000000000014231426641764600141110ustar00rootroot00000000000000syntax: regexp # The recurrent (^|/) idiom in the regexps below should be understood # to mean "at any directory" while the ^ idiom means "from the # project's top-level directory". (^|/).*\.dvi$ (^|/).*\.pdf$ (^|/).*\.o$ (^|/).*\.oct$ (^|/).*\.octlink$ (^|/)octave-core$ (^|/)octave-workspace$ (^|/).*\.tar\.gz$ ## Our Makefile target ^target/ ## Files generated automatically by autoconf and the configure script ^src/aclocal\.m4$ ^src/configure$ ^src/autom4te\.cache($|/) ^src/config\.log$ ^src/config\.status$ ^src/Makefile$ ^src/.*\.m$ # e.g. doc/faq/OctaveFAQ.info # doc/interpreter/octave.info-4 ^doc/.*\.info(-\d)?$ ^doc/\w*/stamp-vti$ ^doc/\w*/version\.texi$ # Emacs tools create these (^|/)TAGS$ (^|/)semantic.cache$ # Other text editors often create these (^|/)~.* bim-1.1.6/COPYING000066400000000000000000000430771426641764600133540ustar00rootroot00000000000000 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 of this license document, but changing it is not allowed. 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If the program is interactive, make it output a short notice like this when it starts in an interactive mode: Gnomovision version 69, Copyright (C) year name of author Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'. This is free software, and you are welcome to redistribute it under certain conditions; type `show c' for details. The hypothetical commands `show w' and `show c' should show the appropriate parts of the General Public License. Of course, the commands you use may be called something other than `show w' and `show c'; they could even be mouse-clicks or menu items--whatever suits your program. You should also get your employer (if you work as a programmer) or your school, if any, to sign a "copyright disclaimer" for the program, if necessary. 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If this is what you want to do, use the GNU Library General Public License instead of this License. bim-1.1.6/DESCRIPTION000066400000000000000000000005661426641764600140230ustar00rootroot00000000000000Name: bim Version: 1.1.6 Date: 2022-07-22 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-1.1.6/INDEX000066400000000000000000000022661426641764600131060ustar00rootroot00000000000000BIM >> 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-1.1.6/NEWS000066400000000000000000000036431426641764600130130ustar00rootroot00000000000000Summary of important user-visible changes for bim 1.1.6: ------------------------------------------------------------------- ** Avoid complex conjugate when changing array shape in bim2a_reaction and bim2a_rhs. 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-1.1.6/doc/000077500000000000000000000000001426641764600130535ustar00rootroot00000000000000bim-1.1.6/doc/fiume.geo000066400000000000000000000011061426641764600146520ustar00rootroot00000000000000Point (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-1.1.6/doc/tutorial.html000066400000000000000000000060361426641764600156110ustar00rootroot00000000000000

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);

bim-1.1.6/inst/000077500000000000000000000000001426641764600132635ustar00rootroot00000000000000bim-1.1.6/inst/bim1a_advection_diffusion.m000066400000000000000000000077171426641764600205500ustar00rootroot00000000000000## 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-1.1.6/inst/bim1a_advection_upwind.m000066400000000000000000000060751426641764600200640ustar00rootroot00000000000000## 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-1.1.6/inst/bim1a_axisymmetric_advection_diffusion.m000066400000000000000000000151561426641764600233420ustar00rootroot00000000000000## 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-1.1.6/inst/bim1a_axisymmetric_advection_upwind.m000066400000000000000000000065711426641764600226630ustar00rootroot00000000000000## 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-1.1.6/inst/bim1a_axisymmetric_laplacian.m000066400000000000000000000047011426641764600212360ustar00rootroot00000000000000## 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-1.1.6/inst/bim1a_axisymmetric_reaction.m000066400000000000000000000107271426641764600211230ustar00rootroot00000000000000## 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-1.1.6/inst/bim1a_axisymmetric_rhs.m000066400000000000000000000052261426641764600201110ustar00rootroot00000000000000## 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-1.1.6/inst/bim1a_laplacian.m000066400000000000000000000036321426641764600164420ustar00rootroot00000000000000## 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-1.1.6/inst/bim1a_reaction.m000066400000000000000000000051371426641764600163240ustar00rootroot00000000000000## 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-1.1.6/inst/bim1a_rhs.m000066400000000000000000000047511426641764600153150ustar00rootroot00000000000000## 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-1.1.6/inst/bim1c_elem_to_nodes.m000066400000000000000000000054121426641764600173320ustar00rootroot00000000000000## 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-1.1.6/inst/bim1c_norm.m000066400000000000000000000072261426641764600154760ustar00rootroot00000000000000## 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-1.1.6/inst/bim2a_advection_diffusion.m000066400000000000000000000275211426641764600205440ustar00rootroot00000000000000## 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-1.1.6/inst/bim2a_advection_upwind.m000066400000000000000000000075321426641764600200640ustar00rootroot00000000000000## 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-1.1.6/inst/bim2a_axisymmetric_advection_diffusion.m000066400000000000000000000340661426641764600233440ustar00rootroot00000000000000## 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-1.1.6/inst/bim2a_axisymmetric_advection_upwind.m000066400000000000000000000107601426641764600226570ustar00rootroot00000000000000## 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-1.1.6/inst/bim2a_axisymmetric_boundary_mass.m000066400000000000000000000144731426641764600221700ustar00rootroot00000000000000## 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-1.1.6/inst/bim2a_axisymmetric_laplacian.m000066400000000000000000000053241426641764600212410ustar00rootroot00000000000000## 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-1.1.6/inst/bim2a_axisymmetric_reaction.m000066400000000000000000000122021426641764600211120ustar00rootroot00000000000000## 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-1.1.6/inst/bim2a_axisymmetric_rhs.m000066400000000000000000000071441426641764600201130ustar00rootroot00000000000000## 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-1.1.6/inst/bim2a_boundary_mass.m000066400000000000000000000046131426641764600173650ustar00rootroot00000000000000## 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-1.1.6/inst/bim2a_laplacian.m000066400000000000000000000034701426641764600164430ustar00rootroot00000000000000## 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); endfunctionbim-1.1.6/inst/bim2a_reaction.m000066400000000000000000000063311426641764600163220ustar00rootroot00000000000000## 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-1.1.6/inst/bim2a_rhs.m000066400000000000000000000060531426641764600153130ustar00rootroot00000000000000## 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-1.1.6/inst/bim2c_global_flux.m000066400000000000000000000134531426641764600170210ustar00rootroot00000000000000## 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-1.1.6/inst/bim2c_intrp.m000066400000000000000000000041611426641764600156530ustar00rootroot00000000000000## 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-1.1.6/inst/bim2c_mesh_properties.m000066400000000000000000000040701426641764600177260ustar00rootroot00000000000000## 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-1.1.6/inst/bim2c_norm.m000066400000000000000000000107741426641764600155010ustar00rootroot00000000000000## 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-1.1.6/inst/bim2c_pde_gradient.m000066400000000000000000000034321426641764600171440ustar00rootroot00000000000000## 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-1.1.6/inst/bim2c_tri_to_nodes.m000066400000000000000000000052021426641764600172040ustar00rootroot00000000000000## 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-1.1.6/inst/bim2c_unknowns_on_side.m000066400000000000000000000034031426641764600200770ustar00rootroot00000000000000## 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-1.1.6/inst/bim3a_advection_diffusion.m000066400000000000000000000072221426641764600205410ustar00rootroot00000000000000## 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-1.1.6/inst/bim3a_boundary_mass.m000066400000000000000000000047751426641764600173770ustar00rootroot00000000000000## 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-1.1.6/inst/bim3a_laplacian.m000066400000000000000000000067071426641764600164520ustar00rootroot00000000000000## 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-1.1.6/inst/bim3a_osc_advection_diffusion.m000066400000000000000000000266771426641764600214240ustar00rootroot00000000000000## 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-1.1.6/inst/bim3a_osc_laplacian.m000066400000000000000000000073101426641764600173050ustar00rootroot00000000000000## 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-1.1.6/inst/bim3a_reaction.m000066400000000000000000000062401426641764600163220ustar00rootroot00000000000000## 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-1.1.6/inst/bim3a_rhs.m000066400000000000000000000060401426641764600153100ustar00rootroot00000000000000## 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-1.1.6/inst/bim3c_global_flux.m000066400000000000000000000101371426641764600170160ustar00rootroot00000000000000## 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-1.1.6/inst/bim3c_intrp.m000066400000000000000000000042271426641764600156570ustar00rootroot00000000000000## 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-1.1.6/inst/bim3c_mesh_properties.m000066400000000000000000000041571426641764600177350ustar00rootroot00000000000000## 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-1.1.6/inst/bim3c_norm.m000066400000000000000000000115521426641764600154750ustar00rootroot00000000000000## 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-1.1.6/inst/bim3c_pde_gradient.m000066400000000000000000000035261426641764600171510ustar00rootroot00000000000000## 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-1.1.6/inst/bim3c_tri_to_nodes.m000066400000000000000000000052261426641764600172130ustar00rootroot00000000000000## 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-1.1.6/inst/bim3c_unknowns_on_faces.m000066400000000000000000000040351426641764600202370ustar00rootroot00000000000000## 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-1.1.6/inst/bimu_bernoulli.m000066400000000000000000000041371426641764600164550ustar00rootroot00000000000000## 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-1.1.6/inst/bimu_logm.m000066400000000000000000000031211426641764600154100ustar00rootroot00000000000000## 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-1.1.6/inst/private/000077500000000000000000000000001426641764600147355ustar00rootroot00000000000000bim-1.1.6/inst/private/__osc_local_laplacian__.m000066400000000000000000000100451426641764600216510ustar00rootroot00000000000000## 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