StiffAssemblingP1OptV1.m

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function R=StiffAssemblingP1OptV1(nq,nme,q,me,areas)
% function R=StiffAssemblingP1OptV1(nq,nme,q,me,areas)
%   Assembly of the Stiffness Matrix using `P_1`-Lagrange finite elements
%   - OptV1 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=StiffAssemblingP1OptV1(Th.nq,Th.nme,Th.me,Th.q,Th.areas);
%  @endverbatim
% Copyright:
%   See \ref license
Ig=zeros(nme*9,1);Jg=zeros(nme*9,1);
Kg=zeros(nme*9,1);

ii=[1 2 3 1 2 3 1 2 3];
jj=[1 1 1 2 2 2 3 3 3];
kk=1:9;
for k=1:nme
  Me=ElemStiffMatP1(q(:,me(1,k)),q(:,me(2,k)),q(:,me(3,k)),areas(k));
  Ig(kk)=me(ii,k); 
  Jg(kk)=me(jj,k);
  Kg(kk)=Me(:);
  kk=kk+9;
end
R=sparse(Ig,Jg,Kg,nq,nq);