Element Mass Matrix
~~~~~~~~~~~~~~~~~~~~~~

We have 

.. math::
    \mathbb{M}^e(T_k) =\frac{|T_k|}{12}\begin{pmatrix} 2 & 1 & 1\\ 1 & 2 & 1\\ 1 & 1 & 2 \end{pmatrix}

    
Then with :math:`\MAT{K}_g` definition (see Section :ref:`PresentationOptV2-label`) , we obtain
 
.. math::
   \MAT{K}_g(3(i-1)+j,k)=|T_k|\frac{1+\delta_{i,j}}{12}  \quad 1\le i,j \le 3,

We represent in figure :num:`figurebuild2dkgmassoptv2` the corresponding row-wise operations.

.. _figureBuild2dKgMassOptV2:

.. figure::  images/Build2D_Kg_Mass_OptV2.png
       :width: 800px
       :scale: 100%
       :align:   center
     
       Construction of :math:`\mathbb{K}_g` associated to 2d **Mass** matrix in

So the vectorized algorithm for :math:`\mathbb{K}_g` computation is simple and given in Algorithm :num:`elemmassmat2dp1vectoralgo`.
       
.. admonition:: Algorithm :num:`elemmassmat2dp1vectoralgo`

    .. _ElemMassMat2DP1vectoralgo:  
          
    .. figure::  images/ElemMassMat2DP1_algo.png
          :width: 600px
          :scale: 100%
          :align:   center
        
          Vectorized algorithm for :math:`\mathbb{K}_g` associated to 2d **Mass** matrix       
       

.. note::
  .. automodule:: pyOptFEM.FEM2D.elemMatrixVec
      :members: ElemMassMat2DP1Vec
      :noindex:
     
Element Stiffness Matrix
~~~~~~~~~~~~~~~~~~~~~~~~~~~

We have :math:`\forall (\il,\jl)\in\ENS{1}{3}^2`

.. math::
    \mathbb{S}_{\il,\jl}^e(T_k)= |T_k| \DOT{\GRAD\BasisFunc_{\jl}^k}{\GRAD\BasisFunc_{\il}^k}.


Using vectorized algorithm function :math:`\FNametxt{GradientVec2D}`  given in Algorithm :num:`gradientvec2dalgo`, we obtain
the vectorized algorithm :num:`elemstiffmat2dp1vectoralgo` for :math:`\mathbb{K}_g` computation for the **Stiffness** matrix in 2d.
       
.. admonition:: Algorithm :num:`elemstiffmat2dp1vectoralgo`

    .. _ElemStiffMat2DP1vectoralgo:  
          
    .. figure::  images/ElemStiffMat2DP1_algo.png
          :width: 600px
          :scale: 100%
          :align:   center
        
          Vectorized algorithm for :math:`\mathbb{K}_g` associated to 2d **Stiffness** matrix       

.. note::
  .. automodule:: pyOptFEM.FEM2D.elemMatrixVec
    :members: ElemStiffMat2DP1Vec
    :noindex:
        
Element Elastic Stiffness  Matrix
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   

- We define on :math:`T_k` the local *alternate* basis :math:`\mathcal{B}_a^k`  by
   
  .. math::
      \mathcal{B}_a^k=\{\BasisFuncTwoD_1^k,\hdots,\BasisFuncTwoD_6^k\}=\left\{
      \begin{pmatrix}  \BasisFunc_1^k \\ 0 \end{pmatrix},
      \begin{pmatrix}  0  \\ \BasisFunc_1^k \end{pmatrix},
      \begin{pmatrix}  \BasisFunc_2^k \\ 0 \end{pmatrix},
      \begin{pmatrix}  0  \\ \BasisFunc_2^k\end{pmatrix},
      \begin{pmatrix} \BasisFunc_3^k \\ 0 \end{pmatrix},
      \begin{pmatrix} 0  \\ \BasisFunc_3^k \end{pmatrix}
      \right\}
      
  where :math:`\BasisFunc_\il^k=\BasisFunc_{\me(\il,k)}.` With notations of :ref:`Presentation-label`, we have
  :math:`\forall (\il,\jl) \in \ENS{1}{6}^2`
      
  .. math::
    \StiffElasElem_{\il,\jl}(T_k)=
    \int_{T_k} \Odv^t(\BasisFuncTwoD^k_\jl) \mathbb{C}\Odv(\BasisFuncTwoD^k_\il)d\q=
    \int_{T_k} \mathcal{H}(\BasisFuncTwoD^k_\jl,\BasisFuncTwoD^k_\il)(\q)d\q
  
  with, :math:`\forall \vecb{u}=(u_1,u_2)\in\HUnD{\DOM}^2,` :math:`\forall \vecb{v}=(v_1,v_2)\in\HUnD{\DOM}^2,`
  
  .. math::
    \begin{array}{lcl}
    \mathcal{H}(\vecb{u},\vecb{v})&=&{\footnotesize
    \DOT{\begin{pmatrix} \gamma & 0\\ 0 & \mu \end{pmatrix}\GRAD u_1 }{\GRAD v_1}
    +\DOT{\begin{pmatrix} 0 & \lambda\\ \mu & 0 \end{pmatrix}\GRAD u_2 }{\GRAD v_1}}\\
    &+&{\footnotesize\DOT{\begin{pmatrix} 0 & \mu\\ \lambda & 0 \end{pmatrix}\GRAD u_1 }{\GRAD v_2}+
    \DOT{\begin{pmatrix} \mu & 0\\ 0 &\gamma\end{pmatrix}\GRAD u_2 }{\GRAD v_2}}
    \end{array}
    :label: eq_StiffElasHop
    
  For example, we can explicitly compute the first two terms in the first column of :math:`\StiffElasElem(T_k)` which are given by
  
  .. math::
    \begin{array}{lcl}
    \StiffElasElem_{1,1}(T_k)&=&\int_{T_k} \mathcal{H}(\BasisFuncTwoD^k_{1},\BasisFuncTwoD^k_{1})(\q)d\q\\
    &=&\int_{T_k} \mathcal{H}\left(
    \begin{pmatrix}
    \BasisFunc^k_{1}\\
    0
    \end{pmatrix},
    \begin{pmatrix}
    \BasisFunc^k_{1}\\
    0
    \end{pmatrix}
    \right)(\q)d\q\\
    &=&|T_k|
    \DOT{\begin{pmatrix} \gamma & 0\\ 0 & \mu \end{pmatrix}\GRAD \BasisFunc^k_{1} }{\GRAD \BasisFunc^k_{1}}
    =|T_k|\left(\gamma\DP{\BasisFunc^k_{1}}{x}\DP{\BasisFunc^k_{1}}{x}+\mu\DP{\BasisFunc^k_{1}}{y}\DP{\BasisFunc^k_{1}}{y}\right).
    \end{array}
  
  and
  
  .. math::
    \begin{array}{lcl}
    \StiffElasElem_{2,1}(T_k)&=&\int_{T_k} \mathcal{H}(\BasisFuncTwoD^k_{1},\BasisFuncTwoD^k_{2})(\q)d\q\\
    &=&\int_{T_k} \mathcal{H}\left(
    \begin{pmatrix}
    \BasisFunc^k_{1}\\
    0
    \end{pmatrix},
    \begin{pmatrix}
    0\\
    \BasisFunc^k_{1}
    \end{pmatrix}
    \right)(\q)d\q\\
    &=&|T_k|
    \DOT{\begin{pmatrix} 0 & \mu\\ \lambda & 0 \end{pmatrix}\GRAD \BasisFunc^k_{1} }{\GRAD \BasisFunc^k_{1}}
    =|T_k|(\lambda+\mu)\DP{\BasisFunc^k_{1}}{x}\DP{\BasisFunc^k_{1}}{y}.
    \end{array}
    
  Using vectorized algorithm function :math:`\FNametxt{GradientVec2D}`  given in Algorithm :num:`gradientvec2dalgo`, we obtain
  the vectorized algorithm :num:`elemstiffmat2dp1vectoralgo` for :math:`\mathbb{K}_g` computation for the **Elastic Stiffness** matrix in 2d.
        
  .. admonition:: Algorithm :num:`elemstiffelasmatba2dp1vectoralgo`

      .. _ElemStiffElasMatBa2DP1vectoralgo:  
            
      .. figure::  images/ElemStiffElasMatBa2DP1_algo.png
            :width: 600px
            :scale: 100%
            :align:   center
          
            Vectorized algorithm for :math:`\mathbb{K}_g` associated to 2d **Elastic Stiffness** matrix
  
  .. note::
    .. automodule:: pyOptFEM.FEM2D.elemMatrixVec
      :members: ElemStiffElasMatBaVec2DP1
      :noindex:

- We define on :math:`T_k` the local *block* basis :math:`\mathcal{B}_b^k`  by
   
  .. math::
      \mathcal{B}_b^k=\{\BasisFuncTwoDB_1^k,\hdots,\BasisFuncTwoDB_6^k\}=\left\{
      \begin{pmatrix}  \BasisFunc_1^k \\ 0 \end{pmatrix},
      \begin{pmatrix}  \BasisFunc_2^k \\ 0 \end{pmatrix},
      \begin{pmatrix} \BasisFunc_3^k \\ 0 \end{pmatrix},
      \begin{pmatrix}  0  \\ \BasisFunc_1^k \end{pmatrix},
      \begin{pmatrix}  0  \\ \BasisFunc_2^k\end{pmatrix},
      \begin{pmatrix} 0  \\ \BasisFunc_3^k \end{pmatrix}
      \right\}
      
  where :math:`\BasisFunc_\il^k=\BasisFunc_{\me(\il,k)}.` 
    
  For example, using formula :eq:`eq_StiffElasHop`, we can  explicitly compute the first two terms in the first column of :math:`\StiffElasElem(T_k)` which are given by
  
  .. math::
    \begin{array}{lcl}
    \StiffElasElem_{1,1}(T_k)&=&\int_{T_k} \mathcal{H}(\BasisFuncTwoDB^k_{1},\BasisFuncTwoDB^k_{1})(\q)d\q\\
    &=&\int_{T_k} \mathcal{H}\left(
    \begin{pmatrix}
    \BasisFunc^k_{1}\\
    0
    \end{pmatrix},
    \begin{pmatrix}
    \BasisFunc^k_{1}\\
    0
    \end{pmatrix}
    \right)(\q)d\q\\
    &=&|T_k|
    \DOT{\begin{pmatrix} \gamma & 0\\ 0 & \mu \end{pmatrix}\GRAD \BasisFunc^k_{1} }{\GRAD \BasisFunc^k_{1}}
    =|T_k|\left(\gamma\DP{\BasisFunc^k_{1}}{x}\DP{\BasisFunc^k_{1}}{x}+\mu\DP{\BasisFunc^k_{1}}{y}\DP{\BasisFunc^k_{1}}{y}\right).
    \end{array}
  
  and
  
  .. math::
    \begin{array}{lcl}
    \StiffElasElem_{2,1}(T_k)&=&\int_{T_k} \mathcal{H}(\BasisFuncTwoDB^k_{1},\BasisFuncTwoDB^k_{2})(\q)d\q\\
    &=&\int_{T_k} \mathcal{H}\left(
    \begin{pmatrix}
    \BasisFunc^k_{1}\\
    0
    \end{pmatrix},
    \begin{pmatrix}
    \BasisFunc^k_{2}\\
    0
    \end{pmatrix}
    \right)(\q)d\q\\
    &=&|T_k|
    \DOT{\begin{pmatrix} \gamma & 0\\ 0 & \mu \end{pmatrix}\GRAD \BasisFunc^k_{1} }{\GRAD \BasisFunc^k_{2}}
    =|T_k|\left(\gamma\DP{\BasisFunc^k_{1}}{x}\DP{\BasisFunc^k_{2}}{x}+\mu\DP{\BasisFunc^k_{1}}{y}\DP{\BasisFunc^k_{2}}{y} \right).
    \end{array}
    
  Using vectorized algorithm function :math:`\FNametxt{GradientVec2D}`  given in Algorithm :num:`gradientvec2dalgo`, we obtain
  the vectorized algorithm :num:`elemstiffelasmatbb2dp1vectoralgo` for :math:`\mathbb{K}_g` computation for the **Elastic Stiffness** matrix in 2d.
        
  .. admonition:: Algorithm :num:`elemstiffelasmatbb2dp1vectoralgo`

      .. _ElemStiffElasMatBb2DP1vectoralgo:  
            
      .. figure::  images/ElemStiffElasMatBb2DP1_algo.png
            :width: 600px
            :scale: 100%
            :align:   center
          
            Vectorized algorithm for :math:`\mathbb{K}_g` associated to 2d **Elastic Stiffness** matrix

  .. note::
    .. automodule:: pyOptFEM.FEM2D.elemMatrixVec
      :members: ElemStiffElasMatBbVec2DP1
      :noindex: