Opt/StiffElasAssembling2DP1OptV1.m

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function K=StiffElasAssembling2DP1OptV1(nq,nme,q,me,areas,lambda,mu,Num)
% function K=StiffElasAssembling2DP1OptV1(nq,nme,q,me,areas,lambda,mu,Num)
%   Assembly of the Stiffness Elasticity Matrix by `P_1`-Lagrange finite elements
%   using "OptV1" version (see report).
%
%   The Stiffness Elasticity Matrix is given by 
%   ``\StiffElas_{m,l}=\int_{\DOMH} \Odv^t(\BasisFuncTwoD_m) \Ocv(\BasisFuncTwoD_l)dT, \ \forall (m,l)\in\ENS{1}{2\,\nq}^2,``
%   where `\BasisFuncTwoD_m` are `P_1`-Lagrange vector basis functions.
%   Here `\Ocv=(\Occ_{xx},\Occ_{yy},\Occ_{xy})^t` and `\Odv=(\Odc_{xx},\Odc_{yy},2\Odc_{xy})^t`
%   are the elastic stress and strain tensors respectively. 
%
% Parameters:
%  nq: total number of nodes in 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`, `\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.
%  lambda: the first Lame coefficient in Hooke's law
%  mu: the second Lame coefficient in Hooke's law
%  Num: 
%    - 0 global alternate numbering with local alternate numbering (classical method), 
%    - 1 global block numbering with local alternate numbering,
%    - 2 global alternate numbering with local block numbering,
%    - 3 global block numbering with local block numbering.
%
% Return values:
%  K: `2\nq\times 2\nq` stiffness elasticity sparse matrix
%
% Example:
%  @verbatim 
%    Th=SquareMesh(10);
%    lambda=1; mu=1;
%    Num = 0; 
%    KK=StiffElasAssembling2DP1OptV1(Th.nq,Th.nme,Th.q,Th.me,Th.areas,lambda,mu,Num);@endverbatim
%  
% See also:
%   #BuildIkFunc, #BuildElemStiffElasMatFunc
% Copyright:
%   See \ref license
ElemStiffElasMat=BuildElemStiffElasMatFunc(Num);
C=[lambda+2*mu,lambda,0;lambda,lambda+2*mu,0;0,0,mu];
GetI=BuildIkFunc(Num,nq);
Ig=zeros(36*nme,1);Jg=zeros(36*nme,1);Kg=zeros(36*nme,1);
kk=1:36;
for k=1:nme
  Me=ElemStiffElasMat(q(:,me(:,k)),areas(k),C);
  I=GetI(me,k);
  jA=ones(6,1)*I;
  iA=jA';

  Ig(kk)=iA(:);
  Jg(kk)=jA(:);
  Kg(kk)=Me(:);
  kk=kk+36;
end
K=sparse(Ig,Jg,Kg,2*nq,2*nq);