| OptFEM2DP1 Toolbox V1.2b2 Matlab/Octave Optimized P1-Lagrange Finite Element Method in 2D

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StiffElasAssemblingP1OptV0.m
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1 function K=StiffElasAssemblingP1OptV0(nq,nme,q,me,areas,lambda,mu,Num)
2 % function K=StiffElasAssemblingP1OptV0(nq,nme,q,me,areas,lambda,mu,Num)
3 % Assembly of the Stiffness Elasticity Matrix by `P_1`-Lagrange finite elements
4 % using "OptV0" version (see report).
5 %
6 % The Stiffness Elasticity Matrix is given by
7 % ``\StiffElas_{m,l}=\int_{\DOMH} \Odv^t(\BasisFuncTwoD_m) \Ocv(\BasisFuncTwoD_l)dT, \ \forall (m,l)\in\ENS{1}{2\,\nq}^2,``
8 % where `\BasisFuncTwoD_m` are `P_1`-Lagrange vector basis functions.
9 % Here `\Ocv=(\Occ_{xx},\Occ_{yy},\Occ_{xy})^t` and `\Odv=(\Odc_{xx},\Odc_{yy},2\Odc_{xy})^t`
10 % are the elastic stress and strain tensors respectively.
11 %
12 % Parameters:
13 % nq: total number of nodes in the mesh, also denoted by `\nq`.
14 % nme: total number of triangles, also denoted by `\nme`.
15 % q: Array of vertices coordinates, `2\times\nq` array. <br/>
16 % `{\q}(\il,j)` is the
17 % `\il`-th coordinate of the `j`-th vertex, `\il\in\{1,2\}` and
18 % `j\in\ENS{1}{\nq}`
19 % me: Connectivity array, `3\times\nme` array. <br/>
20 % `\me(\jl,k)` is the storage index of the
21 % `\jl`-th vertex of the `k`-th triangle in the array `\q`, `\jl\in\{1,2,3\}` and
22 % `k\in{\ENS{1}{\nme}}`.
23 % areas: Array of areas, `1\times\nme` array. areas(k) is the area of the `k`-th triangle.
24 % lambda: the first Lame coefficient in Hooke's law
25 % mu: the second Lame coefficient in Hooke's law
26 % Num:
27 % - 0 global alternate numbering with local alternate numbering (classical method),
28 % - 1 global block numbering with local alternate numbering,
29 % - 2 global alternate numbering with local block numbering,
30 % - 3 global block numbering with local block numbering.
31 %
32 % Return values:
33 % K: `2\nq\times 2\nq` stiffness elasticity sparse matrix
34 %
35 % Example:
36 % @verbatim
37 % Th=SquareMesh(10);
38 % lambda=1; mu=1;
39 % Num = 0;
40 % KK=StiffElasAssemblingP1OptV0(Th.nq,Th.nme,Th.q,Th.me,Th.areas,lambda,mu,Num);@endverbatim
41 %
42 % See also:
43 % #BuildIkFunc, #BuildElemStiffElasMatOptV0Func
44 % Copyright:
45 % See \ref license
46 K=sparse(2*nq,2*nq);
47 GetI=BuildIkFunc(Num,nq);
48 ElemStiffElasMat=BuildElemStiffElasMatOptV0Func(Num);
49 C=[lambda+2*mu,lambda,0;lambda,lambda+2*mu,0;0,0,mu];
50 for k=1:nme
51 I=GetI(me,k);
52 K(I,I)=K(I,I)+ElemStiffElasMat(q(:,me(:,k)),areas(k),C);
53 end