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Uscharfettergummel3.m

function S = Uscharfettergummel3(mesh,alpha,gamma,eta,beta)

  ## -*- texinfo -*-
  ##
  ## @deftypefn {Function File} @
  ## {@var{S}} = Uscharfettergummel3 (@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 
  ## @ifinfo
  ## -div (@var{alpha} * @var{gamma} (@var{eta} grad u - @var{beta} u )) = f
  ## @end ifinfo
  ## 
  ## where: 
  ## @itemize @minus
  ## @item @var{alpha} is an element-wise constant scalar function
  ## @item @var{eta}, @var{gamma} are piecewise linear conforming 
  ## scalar functions
  ## @item @var{beta}  is an 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.p,mesh.e,mesh.t] = Ustructmesh([0:1/3:1],[0:1/3:1],1,1:4);
  ## mesh = Umeshproperties(mesh);
  ## x = mesh.p(1,:)';
  ## Dnodes = Unodesonside(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      = Ucompconst(mesh,ones(Nnodes,1),ones(Nelements,1));
  ## S = Uscharfettergummel3(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{Ucomplap, Ucompconst, Ucompmass2, Uscharfettergummel}
  ## @end deftypefn


  %% This file is part of 
  %%
  %%            SECS2D - A 2-D Drift--Diffusion Semiconductor Device Simulator
  %%         -------------------------------------------------------------------
  %%            Copyright (C) 2004-2006  Carlo de Falco
  %%
  %%
  %%
  %%  SECS2D 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.
  %%
  %%  SECS2D 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 SECS2D; If not, see <http://www.gnu.org/licenses/>.


  Nnodes = columns(mesh.p);
  Nelements = columns(mesh.t);
  
  alphaareak   = reshape (alpha.*sum( mesh.wjacdet,1)',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;
    end
  end       
  
  
  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,:);
  end
  
  etaloc = eta(mesh.t(1:3,:));
  
  eta12    = etaloc(2,:)-etaloc(1,:);
  eta23    = etaloc(3,:)-etaloc(2,:);
  eta31    = etaloc(1,:)-etaloc(3,:);
  
  etalocm1 = Utemplogm(etaloc(2,:),etaloc(3,:));
  etalocm2 = Utemplogm(etaloc(3,:),etaloc(1,:));
  etalocm3 = Utemplogm(etaloc(1,:),etaloc(2,:));
  
  gammaloc = gamma(mesh.t(1:3,:));
  geloc      = gammaloc.*etaloc;
  
  gelocm1 = Utemplogm(geloc(2,:),geloc(3,:));
  gelocm2 = Utemplogm(geloc(3,:),geloc(1,:));
  gelocm3 = Utemplogm(geloc(1,:),geloc(2,:));
  
  [bp12,bm12] = Ubern( (v12 - eta12)./etalocm3);
  [bp23,bm23] = Ubern( (v23 - eta23)./etalocm1);
  [bp31,bm31] = Ubern( (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);
  
  S = sparse(ginode(:),gjnode(:),Sloc(:));

endfunction
  
%!test 
%! [mesh.p,mesh.e,mesh.t] = Ustructmesh([0:1/3:1],[0:1/3:1],1,1:4);
%! mesh = Umeshproperties(mesh);
%! x = mesh.p(1,:)';
%! Dnodes = Unodesonside(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   = ;
%! gamma  = ones(Nnodes,1);
%! f      = Ucompconst(mesh,ones(Nnodes,1),ones(Nelements,1));
%! S = Uscharfettergummel3(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.p,mesh.e,mesh.t] = Ustructmesh([0:1/3:1],[0:1/3:1],1,1:4);
%! mesh = Umeshproperties(mesh);
%! x = mesh.p(1,:)';
%! Dnodes = Unodesonside(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      = Ucompconst(mesh,ones(Nnodes,1),ones(Nelements,1));
%! S = Uscharfettergummel3(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.p,mesh.e,mesh.t] = Ustructmesh([0:1/3:1],[0:1/3:1],1,1:4);
%! mesh = Umeshproperties(mesh);
%! x = mesh.p(1,:)';
%! Dnodes = Unodesonside(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      = Ucompconst(mesh,ones(Nnodes,1),ones(Nelements,1));
%! S = Uscharfettergummel3(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.p,mesh.e,mesh.t] = Ustructmesh([0:1/3:1],[0:1/3:1],1,1:4);
%! mesh = Umeshproperties(mesh);
%! x = mesh.p(1,:)';
%! Dnodes = Unodesonside(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      = Ucompconst(mesh,ones(Nnodes,1),ones(Nelements,1));
%! S = Uscharfettergummel3(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.p,mesh.e,mesh.t] = Ustructmesh([0:1/1e3:1],[0:1/2:1],1,1:4);
%! mesh = Umeshproperties(mesh);
%! x = mesh.p(1,:)';
%! Dnodes = Unodesonside(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   = ;
%! gamma  = 2*x;
%! ff     = 2*(6*x.^2+6*x) - (6*x+6).*(1-2*x)+6*(x-x.^2);
%! f      = Ucompconst(mesh,ff,ones(Nelements,1));
%! S = Uscharfettergummel3(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.p,mesh.e,mesh.t] = Ustructmesh([0:1/1e3:1],[0:1/2:1],1,1:4);
%! mesh = Umeshproperties(mesh);
%! x = mesh.p(1,:)';
%! Dnodes = Unodesonside(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      = Ucompconst(mesh,ff,ones(Nelements,1));
%! S = Uscharfettergummel3(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.p,mesh.e,mesh.t] = Ustructmesh([0:.1:1],[0:.1:1],1,1:4);
%! mesh = Umeshproperties(mesh);
%! x = mesh.p(1,:)';y = mesh.p(2,:)';
%! Dnodes = Unodesonside(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   = Ucompconst(mesh,ff,ones(Nelements,1));
%! S = Uscharfettergummel3(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)


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