MassWAssemblingP1OptV2.m

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function M=MassWAssemblingP1OptV2(nq,nme,me,areas,Tw)
% function M=MassWAssemblingP1OptV2(nq,nme,me,areas,Tw)
%   Assembling Mass Weight Matrix by `P_1`-Lagrange finite elements
%   using "OptV2" version (see report).
%
%   The Mass Weight Matrix is given by 
%   ``\MasseF{w}_{i,j}=\int_\DOMH w(\q)\FoncBase_i(\q) \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`,
%  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.
%  Tw: `\nme` 'double' array, `Tw(i)=w(\q^i),` `\forall i\in\ENS{1}{\nq}`.
%
% Return values:
%  M: `\nq\times\nq` sparse matrix
%
% Example:
%  @verbatim 
%    Th=SquareMesh(10);
%    w=@(x,y) cos(x+y);
%    Tw=w(Th.q(1,:),Th.q(2,:));
%    Mw=MassWAssemblingP1OptV2(Th.nq,Th.nme,Th.me,Th.areas,Tw);
%  @endverbatim
%
% See also:
%   #SquareMesh
%
%% @author Francois Cuvelier @date 2012-03-09
%
% 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
Ig = me([1 2 3 1 2 3 1 2 3],:);
Jg = me([1 1 1 2 2 2 3 3 3],:);
W=Tw(me).*(ones(3,1)*areas/30);
Kg=zeros(9,length(areas));
Kg(1,:)=3*W(1,:)+W(2,:)+W(3,:);
Kg(2,:)=W(1,:)+W(2,:)+W(3,:)/2;
Kg(3,:)=W(1,:)+W(2,:)/2+W(3,:);
Kg(5,:)=W(1,:)+3*W(2,:)+W(3,:);
Kg(6,:)=W(1,:)/2+W(2,:)+W(3,:);
Kg(9,:)=W(1,:)+W(2,:)+3*W(3,:);
Kg([4, 7, 8],:)=Kg([2, 3, 6],:);
M = sparse(Ig,Jg,Kg,nq,nq);