User:Eml5526.s11.team01.roark/Mtg31

[[media: Fe1.s11.mtg31.djvu| Mtg 31]]: Fri, 19 Mar 11 [[media: Fe1.s11.mtg31.djvu| Page 31-1]] Element viewpoint: Element Matices

Instead of using [[media: Fe1.s11.mtg30.djvu|(2) p. 30-4]], [[media: Fe1.s11.mtg30.djvu|(1) p. 30-5]], consider
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$$\displaystyle \tilde{\mathbf K}=\left[ {{K}_{ij}} \right],\underbrace_{\left( n+1 \right)\times \left( n+1 \right)}=\underset{e=1}{\overset{nel}{\mathop \sum }}\,{{\tilde{\mathbf K}}^{e}},{{\underbrace_{\left( n+1 \right)\times \left( n+1 \right)}}^{e}}=\left[ K_{ij}^{e} \right]$$ Recall: “Tilde” means include all degrees of freedom (E (essential)+F (free))
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End Element Viewpoint Section


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$$\displaystyle \tilde{\mathbf K}_{\left( n+1 \right)\times (n+1)}^{e}=L_{(n+1)\times {{n}_{e}}}^k_{{{n}_{e}}\times {{n}_{e}}}^{e}L_{{{n}_{e}}\times (n+1)}^{e}$$
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Where $$\displaystyle {{n}_{e}}$$ = number of degrees of freedom per element and $$\displaystyle {{\mathbf L}^{e}}$$ = “Location” or “scatter” matrix. From
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$$\displaystyle \mathbf{d}_{{{n}_{e}}\times 1}^{e}=\underbrace{\left\{ d_{i}^{e} \right\}}_{\left( 5 \right)p.30-3}=\mathbf{L}_{{{n}_{e}}\times (n+1)}^{e}\tilde{\mathbf d}\underbrace{_{(n+1)\times 1}}_{\left( 4 \right)p.22-2}$$
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[[media: Fe1.s11.mtg31.djvu| Page 31-2]] Example:

[[media: Fe1.s11.mtg30.djvu| p. 30-2]], ID QLEBF -> $$\displaystyle {{n}_{e}}=3$$


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$$\displaystyle {{\mathbf d}^{(1)}}=\left[ \begin{matrix} d_{1}^{(1)} \\ d_{2}^{(1)} \\ d_{3}^{(1)} \\ \end{matrix} \right],{{ \mathbf d}^{(2)}}=\left[ \begin{matrix} d_{1}^{(2)} \\ d_{2}^{(2)} \\ d_{3}^{(2)} \\ \end{matrix} \right]{{ \mathbf d}^{(3)}}=\left[ \begin{matrix} d_{1}^{(3)} \\ d_{2}^{(3)} \\ d_{3}^{(3)} \\ \end{matrix} \right]$$
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$$\displaystyle \mathbf L_{3\times 7}^{(2)}=\left[ \begin{matrix} 0 \\   0  \\   0  \\ \end{matrix}\begin{matrix} 0 \\   0  \\   0  \\ \end{matrix}\begin{matrix} 1 \\   0  \\   0  \\ \end{matrix}\begin{matrix} 0 \\   1  \\   0  \\ \end{matrix}\begin{matrix} 0 \\   0  \\   1  \\ \end{matrix}\begin{matrix} 0 \\   0  \\   0  \\ \end{matrix}\begin{matrix} 0 \\   0  \\   0  \\ \end{matrix} \right]$$ End Section
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[[media: Fe1.s11.mtg31.djvu| Page 31-3]] $$\displaystyle {{\mathbf k}^{e}}$$=Element stiffness / conductance matrix
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$$\displaystyle {{\mathbf k}^{e}}=\left[ k_{ij}^{e} \right],k_{ij}^{e}=\underset{\overset – {\mathop \int }}\,b{{_{i}^{e}}^{\prime }}{{a}_{2}}b{{_{j}^{e}}^{\prime }}dx,i,j=1,2,\ldots ,{{n}_{e}}$$
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In general, $$\displaystyle n={{n}_{F}}$$= number of free degrees of freedom
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$$\displaystyle {{n}_{E}}$$= no. of prescribed degrees of freedom on $$\displaystyle {{\text{ }\!\!\Gamma\!\!\text{ }}_{g}}$$ (essential boundary conditions)
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$$\displaystyle ~n={{n}_{E}}+{{n}_{F}}$$= total number of degrees of freedom
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$$\displaystyle \tilde{\mathbf K}=\left[ {{K}_{ij}} \right],\underbrace_{\tilde{n}\times \tilde{n}}=\underset{e=1}{\overset{nel}{\mathop \sum }}\,{{\tilde{\mathbf K}}^{e}},{{\underbrace_{\tilde{n}\times \tilde{n}}}^{e}}=\left[ K_{ij}^{e} \right]$$
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$$\displaystyle \tilde{\mathbf K}_{\tilde{n}\times \tilde{n}}^{e}=\mathbf L_{\tilde{n}\times {{n}_{e}}}^\mathbf k_{{{n}_{e}}\times {{n}_{e}}}^{e}\mathbf L_{{{n}_{e}}\times \tilde{n}}^{e}$$
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[[media: Fe1.s11.mtg31.djvu| Page 31-3]]

HW 6.1:

Similar to HW 5.{1,3,7}, but using QLEBF with uniform discretization (equidistant element nodes) nel=2,4,6,7,… End HW 6.1
 * 1) For nel=2, compare $$\displaystyle \tilde{\mathbf K}=\underset{e=1}{\overset{2}{\mathop \sum }}\,{{\tilde{\mathbf K}}^{e}}$$with $$\displaystyle {{\tilde{\mathbf K}}^{e}}$$ by [[media: Fe1.s11.mtg30.djvu|(1) p. 30-5]], display $$\displaystyle {{\tilde{\mathbf K}}^{e}}$$, e=1,2.
 * 2) Compare $$\displaystyle {{\tilde{\mathbf k}}^{e}}$$, $$\displaystyle {{\mathbf L}^{e}}$$for e=1,2.
 * 3) Compare $$\displaystyle {{\mathbf K}^{e}}={{\mathbf L}^}{{\mathbf k}^{e}}{{\mathbf L}^{e}}$$for e=1,2. Compare to 1)
 * 4) Plot all QLEBF for nel=3.
 * 5) Plot $$\displaystyle u_^{h}$$ vs $$\displaystyle u,\left[ u_^{h}\left( 0.5 \right)-u(0.5) \right]$$ vs $$\displaystyle \tilde{n}$$.

HW 6.2:

Similar to HW 5.{2,4,8}, but using QLEBF with uniform discretization (equidistant element nodes) nel=2,4,6,8,… Same tasks as in HW 6.1

End HW 6.2