StiffAssemblingP1OptV2.m

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function R=StiffAssemblingP1OptV2(nq,nme,q,me,areas)
% function R=StiffAssemblingP1OptV2(nq,nme,q,me,areas)
%   Assembly of the Stiffness Matrix using `P_1`-Lagrange finite elements
%   - OptV2 version (see report).
%
%   The Stiffness Matrix `\Stiff` 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 denoted by `\nq`.
%  nme: total number of triangles, also denoted by `\nme`.
%  q: Array of vertices coordinates, `2\times\nq` array. <br/>
%  `{\q}(\il,j)` is the
%  `\il`-th coordinate of the `j`-th vertex, `\il\in\{1,2\}` and
%  `j\in\ENS{1}{\nq}`
%  me: Connectivity array, `3\times\nme` array.<br/>
%  `\me(\jl,k)` is the storage index of the
%  `\jl`-th  vertex of the `k`-th triangle in the array `\q` of vertices coordinates, `\jl\in\{1,2,3\}` and
%       `k\in{\ENS{1}{\nme}}`.
%  areas: Array of areas, `1\times\nme` array. areas(k) is the area of the `k`-th triangle.
%
% Return values:
%  R: Stiffness Matrix, `\nq\times\nq` sparse matrix.
%
% Example:
%  @verbatim 
%    Th=SquareMesh(10);
%    R=StiffAssemblingP1OptV2(Th.nq,Th.nme,Th.me,Th.q,Th.areas);
%  @endverbatim
% Copyright:
%   See \ref license
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);