StiffAssemblingP1OptV2.m

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function R=StiffAssemblingP1OptV2(nq,nme,q,me,areas)
% function R=StiffAssemblingP1OptV2(nq,nme,q,me,areas)
%   Assembling Stiff Matrix by `P_1`-Lagrange finite elements
%   using "OptV2" version (see report).
%
%   The Stiff Matrix is given by 
%   ``\Stiff_{i,j}=\int_\DOMH \DOT{\GRAD\FoncBase_i(\q)}{\GRAD\FoncBase_j(\q)}d\q,\ \forall (i,j)\in\ENS{1}{\nq}^2``
%   where `\FoncBase_i` are `P_1`-Lagrange basis functions.
%
% Parameters:
%  nq: total number of nodes of the mesh, also noted `\nq`.
%  nme: number of triangles, also noted `\nme`.
%  'q': `2\times\nq` array, `{\q}(\il,j)` is the  `\il`-th coordinate of the `j`-th vertex, `\il\in\{1,2\}` and `j\in\{1,\hdots,n_q\}.`
%       Also noted `\q`.
%  me: `3\times\nme` 'int32' array,`\me(\jl,k)` index of storage, in the array `\q`, of the `\jl`-th
%       vertex of the triangle of index `k`, `\jl\in\{1,2,3\}` and `k\in\{1,\hdots,\nme\}.`
%       Also noted `\me`.
%  areas: `1\times\nme` array, areas(k) is the area of triangle k.
%
% Return values:
%  R: `\nq\times\nq` sparse matrix
%
% Example:
%  @verbatim 
%    Th=SquareMesh(10);
%    R=StiffAssemblingP1OptV2(Th.nq,Th.nme,Th.me,Th.q,Th.areas);@endverbatim
%  
% See also:
%   #SquareMesh
% @author Francois Cuvelier @date 2011-11-18
%
% @new{1,1,fc,2012-11-22} Added doxygen documentation using mtoc++ tool
%
% Copyright (c) 2012, Francois Cuvelier, Gilles Scarella
% All rights reserved.
%
% Redistribution and use in source and binary forms, with or without
% modification, are permitted only in compliance with the BSD license, see
% http://www.opensource.org/licenses/bsd-license.php
me=double(me);
Ig = me([1 2 3 1 2 3 1 2 3],:);
Jg = me([1 1 1 2 2 2 3 3 3],:);

q1 =q(:,me(1,:)); q2 =q(:,me(2,:)); q3 =q(:,me(3,:));
u = q2-q3; v=q3-q1; w=q1-q2;
clear q1 q2 q3
areas4=4*areas;
Kg=zeros(9,nme);
Kg(1,:)=sum(u.*u,1)./areas4; % K1 ou G11
Kg(2,:)=sum(v.*u,1)./areas4; % K2 ou G12
Kg(3,:)=sum(w.*u,1)./areas4; % K3 ou G13
Kg(5,:)=sum(v.*v,1)./areas4; % K5 ou G22
Kg(6,:)=sum(w.*v,1)./areas4; % K6 ou G23
Kg(9,:)=sum(w.*w,1)./areas4; % K9 ou G33
Kg([4, 7, 8],:)=Kg([2, 3, 6],:);
R = sparse(Ig,Jg,Kg,nq,nq);