Element Mass Matrix¶

We have

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

Then with $\MAT{K}_g$ definition (see Section New Optimized assembly algorithm (OptV2 version)) , we obtain

$\MAT{K}_g(3(i-1)+j,k)=|T_k|\frac{1+\delta_{i,j}}{12} \quad 1\le i,j \le 3,$

We represent in figure 13 the corresponding row-wise operations.

Figure 13: Construction of $\mathbb{K}_g$ associated to 2d Mass matrix in

So the vectorized algorithm for $\mathbb{K}_g$ computation is simple and given in Algorithm 14.

Algorithm 14

Figure 14: Vectorized algorithm for $\mathbb{K}_g$ associated to 2d Mass matrix

Note

pyOptFEM.FEM2D.elemMatrixVec.ElemMassMat2DP1Vec(areas)[source]

Computes all the element Mass matrices $\mathbb{M}^e(T_k)$ for $k\in\{0,\hdots,\nme-1\}$

Parameters:	areas ( $\nme$ numpy array of floats) – areas of all the mesh elements.
Returns:	a one dimensional numpy array of size $9 \nme$

Element Stiffness Matrix¶

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

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

Using vectorized algorithm function $\FNametxt{GradientVec2D}$ given in Algorithm 12, we obtain the vectorized algorithm 15 for $\mathbb{K}_g$ computation for the Stiffness matrix in 2d.

Algorithm 15

Figure 15: Vectorized algorithm for $\mathbb{K}_g$ associated to 2d Stiffness matrix

Note

pyOptFEM.FEM2D.elemMatrixVec.ElemStiffMat2DP1Vec(nme, q, me, areas)[source]

Computes all the element stiffness matrices $\mathbb{S}^e(T_k)$ for $k\in\{0,\hdots,\nme-1\}$

Parameters:	nme (int) – number of mesh elements, q ( $\nq\times 2$ numpy array of floats) – mesh vertices, me ( $\nme\times 3$ numpy array of integers) – mesh connectivity, areas ( $\nme$ numpy array of floats) – areas of all the mesh elements.
Returns:	a one dimensional numpy array of size $9 \nme$

Element Elastic Stiffness Matrix¶

We define on $T_k$ the local alternate basis $\mathcal{B}_a^k$ by

$\mathcal{B}_a^k=\{\BasisFuncTwoD_1^k,\hdots,\BasisFuncTwoD_6^k\}=\left\{ \begin{pmatrix} \BasisFunc_1^k \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ \BasisFunc_1^k \end{pmatrix}, \begin{pmatrix} \BasisFunc_2^k \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ \BasisFunc_2^k\end{pmatrix}, \begin{pmatrix} \BasisFunc_3^k \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ \BasisFunc_3^k \end{pmatrix} \right\}$

where $\BasisFunc_\il^k=\BasisFunc_{\me(\il,k)}.$ With notations of Presentation, we have $\forall (\il,\jl) \in \ENS{1}{6}^2$

$\StiffElasElem_{\il,\jl}(T_k)= \int_{T_k} \Odv^t(\BasisFuncTwoD^k_\jl) \mathbb{C}\Odv(\BasisFuncTwoD^k_\il)d\q= \int_{T_k} \mathcal{H}(\BasisFuncTwoD^k_\jl,\BasisFuncTwoD^k_\il)(\q)d\q$

with, $\forall \vecb{u}=(u_1,u_2)\in\HUnD{\DOM}^2,$ $\forall \vecb{v}=(v_1,v_2)\in\HUnD{\DOM}^2,$

(1) $\begin{array}{lcl} \mathcal{H}(\vecb{u},\vecb{v})&=&{\footnotesize \DOT{\begin{pmatrix} \gamma & 0\\ 0 & \mu \end{pmatrix}\GRAD u_1 }{\GRAD v_1} +\DOT{\begin{pmatrix} 0 & \lambda\\ \mu & 0 \end{pmatrix}\GRAD u_2 }{\GRAD v_1}}\\ &+&{\footnotesize\DOT{\begin{pmatrix} 0 & \mu\\ \lambda & 0 \end{pmatrix}\GRAD u_1 }{\GRAD v_2}+ \DOT{\begin{pmatrix} \mu & 0\\ 0 &\gamma\end{pmatrix}\GRAD u_2 }{\GRAD v_2}} \end{array}$

For example, we can explicitly compute the first two terms in the first column of $\StiffElasElem(T_k)$ which are given by

$\begin{array}{lcl} \StiffElasElem_{1,1}(T_k)&=&\int_{T_k} \mathcal{H}(\BasisFuncTwoD^k_{1},\BasisFuncTwoD^k_{1})(\q)d\q\\ &=&\int_{T_k} \mathcal{H}\left( \begin{pmatrix} \BasisFunc^k_{1}\\ 0 \end{pmatrix}, \begin{pmatrix} \BasisFunc^k_{1}\\ 0 \end{pmatrix} \right)(\q)d\q\\ &=&|T_k| \DOT{\begin{pmatrix} \gamma & 0\\ 0 & \mu \end{pmatrix}\GRAD \BasisFunc^k_{1} }{\GRAD \BasisFunc^k_{1}} =|T_k|\left(\gamma\DP{\BasisFunc^k_{1}}{x}\DP{\BasisFunc^k_{1}}{x}+\mu\DP{\BasisFunc^k_{1}}{y}\DP{\BasisFunc^k_{1}}{y}\right). \end{array}$

and

$\begin{array}{lcl} \StiffElasElem_{2,1}(T_k)&=&\int_{T_k} \mathcal{H}(\BasisFuncTwoD^k_{1},\BasisFuncTwoD^k_{2})(\q)d\q\\ &=&\int_{T_k} \mathcal{H}\left( \begin{pmatrix} \BasisFunc^k_{1}\\ 0 \end{pmatrix}, \begin{pmatrix} 0\\ \BasisFunc^k_{1} \end{pmatrix} \right)(\q)d\q\\ &=&|T_k| \DOT{\begin{pmatrix} 0 & \mu\\ \lambda & 0 \end{pmatrix}\GRAD \BasisFunc^k_{1} }{\GRAD \BasisFunc^k_{1}} =|T_k|(\lambda+\mu)\DP{\BasisFunc^k_{1}}{x}\DP{\BasisFunc^k_{1}}{y}. \end{array}$

Using vectorized algorithm function $\FNametxt{GradientVec2D}$ given in Algorithm 12, we obtain the vectorized algorithm 15 for $\mathbb{K}_g$ computation for the Elastic Stiffness matrix in 2d.

Algorithm 16

Figure 16: Vectorized algorithm for $\mathbb{K}_g$ associated to 2d Elastic Stiffness matrix

Note

pyOptFEM.FEM2D.elemMatrixVec.ElemStiffElasMatBaVec2DP1(nme, q, me, areas, L, M, **kwargs)[source]

Computes all the element elastic stiffness matrices $\mathbb{K}^e(T_k)$ for $k\in\{0,\hdots,\nme-1\}$ in local alternate basis.

Parameters:	nme (int) – number of mesh elements, q (`(2,nq)` numpy array of floats) – mesh vertices, me (`(3,nme)` numpy array of integers) – mesh connectivity, areas (`(nme,)` numpy array of floats) – areas of all the mesh elements. L (float) – the $\lambda$ Lame parameter, M (float) – the $\mu$ Lame parameter.
Returns:	a `(36nme,)` numpy* array of floats.

We define on $T_k$ the local block basis $\mathcal{B}_b^k$ by

$\mathcal{B}_b^k=\{\BasisFuncTwoDB_1^k,\hdots,\BasisFuncTwoDB_6^k\}=\left\{ \begin{pmatrix} \BasisFunc_1^k \\ 0 \end{pmatrix}, \begin{pmatrix} \BasisFunc_2^k \\ 0 \end{pmatrix}, \begin{pmatrix} \BasisFunc_3^k \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ \BasisFunc_1^k \end{pmatrix}, \begin{pmatrix} 0 \\ \BasisFunc_2^k\end{pmatrix}, \begin{pmatrix} 0 \\ \BasisFunc_3^k \end{pmatrix} \right\}$

where $\BasisFunc_\il^k=\BasisFunc_{\me(\il,k)}.$

For example, using formula (1), we can explicitly compute the first two terms in the first column of $\StiffElasElem(T_k)$ which are given by

$\begin{array}{lcl} \StiffElasElem_{1,1}(T_k)&=&\int_{T_k} \mathcal{H}(\BasisFuncTwoDB^k_{1},\BasisFuncTwoDB^k_{1})(\q)d\q\\ &=&\int_{T_k} \mathcal{H}\left( \begin{pmatrix} \BasisFunc^k_{1}\\ 0 \end{pmatrix}, \begin{pmatrix} \BasisFunc^k_{1}\\ 0 \end{pmatrix} \right)(\q)d\q\\ &=&|T_k| \DOT{\begin{pmatrix} \gamma & 0\\ 0 & \mu \end{pmatrix}\GRAD \BasisFunc^k_{1} }{\GRAD \BasisFunc^k_{1}} =|T_k|\left(\gamma\DP{\BasisFunc^k_{1}}{x}\DP{\BasisFunc^k_{1}}{x}+\mu\DP{\BasisFunc^k_{1}}{y}\DP{\BasisFunc^k_{1}}{y}\right). \end{array}$

and

$\begin{array}{lcl} \StiffElasElem_{2,1}(T_k)&=&\int_{T_k} \mathcal{H}(\BasisFuncTwoDB^k_{1},\BasisFuncTwoDB^k_{2})(\q)d\q\\ &=&\int_{T_k} \mathcal{H}\left( \begin{pmatrix} \BasisFunc^k_{1}\\ 0 \end{pmatrix}, \begin{pmatrix} \BasisFunc^k_{2}\\ 0 \end{pmatrix} \right)(\q)d\q\\ &=&|T_k| \DOT{\begin{pmatrix} \gamma & 0\\ 0 & \mu \end{pmatrix}\GRAD \BasisFunc^k_{1} }{\GRAD \BasisFunc^k_{2}} =|T_k|\left(\gamma\DP{\BasisFunc^k_{1}}{x}\DP{\BasisFunc^k_{2}}{x}+\mu\DP{\BasisFunc^k_{1}}{y}\DP{\BasisFunc^k_{2}}{y} \right). \end{array}$

Using vectorized algorithm function $\FNametxt{GradientVec2D}$ given in Algorithm 12, we obtain the vectorized algorithm 17 for $\mathbb{K}_g$ computation for the Elastic Stiffness matrix in 2d.

Algorithm 17

Figure 17: Vectorized algorithm for $\mathbb{K}_g$ associated to 2d Elastic Stiffness matrix

Note

pyOptFEM.FEM2D.elemMatrixVec.ElemStiffElasMatBbVec2DP1(nme, q, me, areas, L, M, **kwargs)[source]

Computes all the element elastic stiffness matrices $\mathbb{K}^e(T_k)$ for $k\in\{0,\hdots,\nme-1\}$ in local block basis.

Parameters:	nme (int) – number of mesh elements, q (`(2,nq)` numpy array of floats) – mesh vertices, me (`(3,nme)` numpy array of integers) – mesh connectivity, areas (`(nme,)` numpy array of floats) – areas of all the mesh elements. L (float) – the $\lambda$ Lame parameter, M (float) – the $\mu$ Lame parameter.
Returns:	a `(36nme,)` numpy* array of floats.

Element Mass Matrix¶

Element Stiffness Matrix¶

Element Elastic Stiffness Matrix¶

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Element Mass Matrix¶

Element Stiffness Matrix¶

Element Elastic Stiffness Matrix¶

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