## Copyright (C) 2007,2008,2009 Carlo de Falco, Massimiliano Culpo ## ## BIM - Box Integration Method Package for Octave ## ## 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 <http://www.gnu.org/licenses/>. ## ## ## AUTHORS: ## ## Carlo de Falco <cdf _AT_ users.sourceforge.net> ## ## Culpo Massimiliano ## Bergische Universitaet Wuppertal ## Fachbereich C - Mathematik und Naturwissenschaften ## Arbeitsgruppe fuer Angewandte MathematD-42119 Wuppertal Gaussstr. 20 ## D-42119 Wuppertal, Germany ## -*- texinfo -*- ## ## @deftypefn {Function File} @ ## {@var{A}} = BIM2Aadvdiff (@var{mesh}, @var{alpha}, @var{gamma}, @var{eta}, @var{beta}) ## ## Builds the Scharfetter-Gummel matrix for the ## discretization of the LHS ## of the equation ## @iftex ## @tex ## $ -div ( \alpha \gamma ( \eta \vect{\nabla} u - \vect{beta} u )) = f $ ## @end tex ## @end iftex ## @ifnottex ## -div (@var{alpha} * @var{gamma} (@var{eta} grad u - @var{beta} u )) = f ## @end ifnottex ## ## where: ## @itemize @minus ## @item @var{alpha}: element-wise constant scalar function ## @item @var{eta}, @var{gamma}: piecewise linear conforming ## scalar functions ## @item @var{beta}: element-wise constant vector function ## @end itemize ## ## 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 = MSH2Mstructmesh([0:1/3:1],[0:1/3:1],1,1:4); ## mesh = BIM2Cmeshproperties(mesh); ## x = mesh.p(1,:)'; ## Dnodes = BIM2Cunknownsonside(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 = BIM2Arhs(mesh,ones(Nnodes,1),ones(Nelements,1)); ## S = BIM2Aadvdiff(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) ## @end example ## ## @seealso{BIM2Arhs, BIM2Areaction, BIM2Cmeshproperties} ## @end deftypefn function [A] = BIM2Aadvdiff(mesh,alpha,gamma,eta,beta) Nnodes = columns(mesh.p); Nelements = columns(mesh.t); alphaareak = reshape (alpha.*mesh.area,1,1,Nelements); shg = mesh.shg(:,:,:); ## Build local Laplacian matrix Lloc = zeros(3,3,Nelements); 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,Nelements]) 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("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 = BIMUlogm(etaloc(2,:),etaloc(3,:)); etalocm2 = BIMUlogm(etaloc(3,:),etaloc(1,:)); etalocm3 = BIMUlogm(etaloc(1,:),etaloc(2,:)); gammaloc = gamma(mesh.t(1:3,:)); geloc = gammaloc.*etaloc; gelocm1 = BIMUlogm(geloc(2,:),geloc(3,:)); gelocm2 = BIMUlogm(geloc(3,:),geloc(1,:)); gelocm3 = BIMUlogm(geloc(1,:),geloc(2,:)); [bp12,bm12] = BIMUbern( (v12 - eta12)./etalocm3); [bp23,bm23] = BIMUbern( (v23 - eta23)./etalocm1); [bp31,bm31] = BIMUbern( (v31 - eta31)./etalocm2); bp12 = reshape(gelocm3.*etalocm3.*bp12,1,1,Nelements).*Lloc(1,2,:); bm12 = reshape(gelocm3.*etalocm3.*bm12,1,1,Nelements).*Lloc(1,2,:); bp23 = reshape(gelocm1.*etalocm1.*bp23,1,1,Nelements).*Lloc(2,3,:); bm23 = reshape(gelocm1.*etalocm1.*bm23,1,1,Nelements).*Lloc(2,3,:); bp31 = reshape(gelocm2.*etalocm2.*bp31,1,1,Nelements).*Lloc(3,1,:); bm31 = reshape(gelocm2.*etalocm2.*bm31,1,1,Nelements).*Lloc(3,1,:); Sloc(1,1,:) = (-bm12-bp31)./reshape(etaloc(1,:),1,1,Nelements); Sloc(1,2,:) = bp12./reshape(etaloc(2,:),1,1,Nelements); Sloc(1,3,:) = bm31./reshape(etaloc(3,:),1,1,Nelements); Sloc(2,1,:) = bm12./reshape(etaloc(1,:),1,1,Nelements); Sloc(2,2,:) = (-bp12-bm23)./reshape(etaloc(2,:),1,1,Nelements); Sloc(2,3,:) = bp23./reshape(etaloc(3,:),1,1,Nelements); Sloc(3,1,:) = bp31./reshape(etaloc(1,:),1,1,Nelements); Sloc(3,2,:) = bm23./reshape(etaloc(2,:),1,1,Nelements); Sloc(3,3,:) = (-bm31-bp23)./reshape(etaloc(3,:),1,1,Nelements); A = sparse(ginode(:),gjnode(:),Sloc(:)); endfunction %!test %! [mesh] = MSH2Mstructmesh([0:1/3:1],[0:1/3:1],1,1:4); %! mesh = BIM2Cmeshproperties(mesh); %! x = mesh.p(1,:)'; %! Dnodes = BIM2Cunknownsonside(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 = BIM2Arhs(mesh,ones(Nnodes,1),ones(Nelements,1)); %! S = BIM2Aadvdiff(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] = MSH2Mstructmesh([0:1/3:1],[0:1/3:1],1,1:4); %! mesh = BIM2Cmeshproperties(mesh); %! x = mesh.p(1,:)'; %! Dnodes = BIM2Cunknownsonside(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 = BIM2Arhs(mesh,ones(Nnodes,1),ones(Nelements,1)); %! S = BIM2Aadvdiff(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] = MSH2Mstructmesh([0:1/3:1],[0:1/3:1],1,1:4); %! mesh = BIM2Cmeshproperties(mesh); %! x = mesh.p(1,:)'; %! Dnodes = BIM2Cunknownsonside(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 = BIM2Arhs(mesh,ones(Nnodes,1),ones(Nelements,1)); %! S = BIM2Aadvdiff(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] = MSH2Mstructmesh([0:1/3:1],[0:1/3:1],1,1:4); %! mesh = BIM2Cmeshproperties(mesh); %! x = mesh.p(1,:)'; %! Dnodes = BIM2Cunknownsonside(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 = BIM2Arhs(mesh,ones(Nnodes,1),ones(Nelements,1)); %! S = BIM2Aadvdiff(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] = MSH2Mstructmesh([0:1/1e3:1],[0:1/2:1],1,1:4); %! mesh = BIM2Cmeshproperties(mesh); %! x = mesh.p(1,:)'; %! Dnodes = BIM2Cunknownsonside(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 = BIM2Arhs(mesh,ff,ones(Nelements,1)); %! S = BIM2Aadvdiff(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] = MSH2Mstructmesh([0:1/1e3:1],[0:1/2:1],1,1:4); %! mesh = BIM2Cmeshproperties(mesh); %! x = mesh.p(1,:)'; %! Dnodes = BIM2Cunknownsonside(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 = BIM2Arhs(mesh,ff,ones(Nelements,1)); %! S = BIM2Aadvdiff(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] = MSH2Mstructmesh([0:.1:1],[0:.1:1],1,1:4); %! mesh = BIM2Cmeshproperties(mesh); %! x = mesh.p(1,:)';y = mesh.p(2,:)'; %! Dnodes = BIM2Cunknownsonside(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 = BIM2Arhs(mesh,ff,ones(Nelements,1)); %! S = BIM2Aadvdiff(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)