User:Eml5526.s11.team5.JA/Mtg22

Mtg 22: Tue, 22 Feb 11

[[media: Fe1.s11.mtg22.djvu| Page 22-1 ]]

[[media: Fe1.s11.mtg21.djvu|  page 21-3 ]]:


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$$ u(\beta) \cong u^h(\beta) = g = \sum_{j= {\color{Red}0}}^n d_j b_j (\beta) \ $$

$$
 *  $$ \displaystyle {\color{Red}(1)}
 * }

, where $$ g \ $$ is essential boundary condition [[media: Fe1.s11.mtg19.djvu| (1) page 19-3 ]]: Constraint at $$ x = \beta \ $$

CBS [[media: Fe1.s11.mtg20.djvu| (1)  page 20-2 ]]: $$ u^h(\beta) = d_0 b_0 (\beta) = g \ $$

Typically, choose

Remaining $$ \left \{ d_j, j= {\color{Red} 1},...., n  \right \} \ $$ are unconstrained unknowns to be solved for. How ?

Define: [[media: Fe1.s11.mtg20.djvu| (1) page 20-3 ]]:


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$$ \mathbf{c}_{\color{Red} E} := \left \{ c_{\color{Red} 0} \right \} = \left \{ {\color{Red} 0}  \right \}_{\color{Red} 1x1} \ $$

$$
 *  $$ \displaystyle {\color{Red}(5)}
 * }

, where subscript $$ {\color{Red} E} \ $$ stands for "essential" (homogeneous)


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$$ \mathbf{c}_{\color{Red} F} := \left \{ c_i, i = {\color{Red} 1},...,n \right \}_{\color{Red} nx1} \ $$

$$
 *  $$ \displaystyle {\color{Red}(6)}
 * }

, where subscript $$ {\color{Red} F} \ $$ stands for "free" (unconstrained),  F&B, p. 21

[[media: Fe1.s11.mtg22.djvu| Page 22-2 ]]


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$$ \tilde{\mathbf{c}} = \left \{ \frac{\mathbf{c}_E} {\mathbf{c}_F} \right \}_{\color{Red} (n+1)x1} = \left \{ \frac {\mathbf{c}_F} \right \}_{\color{Red} (n+1)x1} \ $$

$$
 *  $$ \displaystyle {\color{Red}(1)}
 * }

Similarly:

[[media: Fe1.s11.mtg22.djvu| (3) page 22-1 ]]:


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$$ \mathbf{d}_{\color{Red} E} := \left \{ d_{\color{Red} 0} \right \} = \left \{ {\color{Red} g} \right \}_{\color{Red} 1x1} \ $$

$$
 *  $$ \displaystyle {\color{Red}(2)}
 * }


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$$ \mathbf{d}_{\color{Red} F} := \left \{ d_j, j = {\color{Red} 1},...,n \right \}_{\color{Red} 1x1} \ $$

$$
 *  $$ \displaystyle {\color{Red}(3)}
 * }

, where subscripts $$ {\color{Red} E} \ $$ and $$ {\color{Red} F} \ $$ stands for "Essential" and "Free" (unconstrained), respectively (F&B, p. 21)


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$$ \tilde{\mathbf{d}} = \left \{ \frac{\mathbf{d}_E} {\mathbf{d}_F} \right \}_{\color{Red} (n+1)x1} = \left \{ \frac {\mathbf{d}_F} \right \}_{\color{Red} (n+1)x1} \ $$

$$
 *  $$ \displaystyle {\color{Red}(4)}
 * }

Mass/Capacitance

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$$ \tilde{\mathbf{M}}_{\color{Red} (n+1)x(n+1)} :=

\begin{bmatrix} \mathbf{M}_{EE}   & | & \mathbf{M}_{EF}    \\ -- & - & -- \\ \mathbf{M}_{FE}   & | & \mathbf{M}_{FF} \end{bmatrix}

\ $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(5)}
 * }

<span id="(1)">
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$$ \mathbf{M}_{EE_{\color{Red} 1x1}} = \left [ M_{\color{Red} 00} \right ], M_{\color{Red} 00} = \tilde{M}_{\color{Red} 00} = \tilde{m}(b_{\color{Red} 0}, b_{\color{Red} 0}) \ $$ [[media: Fe1.s11.mtg16.djvu| (3) page 16-2 ]], [[media: Fe1.s11.mtg19.djvu| (2)  page 19-1 ]]

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(6)}
 * }

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$$ \mathbf{M}_{EF_{\color{Red} 1xn}} = \left [ M_{\color{Red}0j}, j = 1,..,n \right ], M_{{\color{Red} 0}j} = \tilde{M}_{{\color{Red} 0}j} = \tilde{m} (b_{\color{Red} 0}, b_{\color{Red} j}) \ $$ [[media: Fe1.s11.mtg16.djvu| (3)  page 16-2 ]], [[media: Fe1.s11.mtg19.djvu| (2)  page 19-1 ]]

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(7)}
 * }

, where $$ {\color{Red}0j} \ $$ in $$ M_{\color{Red}0j} \ $$ represent row  and  column, respectively.

[[media: Fe1.s11.mtg22.djvu| Page 22-3 ]]

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$$ \mathbf{M}_{FE_{\color{Red} nx1}} = \mathbf{M}^{\color{Red} T}_{EF_{\color{Red} nx1}} \ {\color{Blue} Transpose}  \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(1)}
 * }

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$$ \mathbf{M}_{FF_{\color{Red} nxn}} = \left [ M_{\color{Red} ij}, i,j = 1,..., n \right ], M_{ij} = \tilde{M}_{ij} \underbrace{=}_{\color{Red} (2)} \tilde{m}(b_{\color{Red} i}, b_{\color{Red} j}) \ $$ [[media: Fe1.s11.mtg19.djvu| (2) page 19-1 ]]

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(2)}
 * }

, where $$ {\color{Red} i} \ $$ and $$ {\color{Red} j} \ $$ are row  and  column, respectively.

Stiffness/Conductivity matrix
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$$ \tilde{\mathbf{K}}_{\color{Red} (n+1)x(n+1)} :=

\begin{bmatrix} \mathbf{K}_{EE}   & | & \mathbf{K}_{EF}    \\ -- & - & -- \\ \mathbf{K}_{FE}   & | & \mathbf{K}_{FF} \end{bmatrix}

\ $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(3)}
 * }

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$$ \mathbf{K}_{EE_{\color{Red} 1x1}} = \left [ K_{\color{Red} 00} \right ], K_{\color{Red} 00} = \tilde{K}_{\color{Red} 00} = \tilde{k}(b_{\color{Red} 0}, b_{\color{Red} 0}) \ $$ [[media: Fe1.s11.mtg16.djvu| (2) page 16-2 ]], [[media: Fe1.s11.mtg18.djvu| (3)  page 18-2 ]]

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(4)}
 * }

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$$ \mathbf{K}_{EF_{\color{Red} 1xn}} = \left [ K_{\color{Red}0j}, j = 1,..,n \right ], K_{{\color{Red} 0}j} = \tilde{K}_{{\color{Red} 0}j} = \tilde{k} (b_{\color{Red} 0}, b_{\color{Red} j}) \ $$ [[media: Fe1.s11.mtg16.djvu| (2)  page 16-2 ]], [[media: Fe1.s11.mtg19.djvu| (3)  page 18-2 ]]

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(5)}
 * }

, where $$ {\color{Red}0j} \ $$ in $$ M_{\color{Red}0j} \ $$ represent row  and  column, respectively.

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$$ \mathbf{K}_{FE_{\color{Red}nx1}} = \mathbf{K}^{\color{Red}T}_{EF_{\color{Red}nx1}} \ {\color{Blue}(Transpose)} \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle
 * }

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$$ \mathbf{K}_{FF_{\color{Red} nxn}} = \left [ K_{\color{Red} ij}, i,j = 1,..., n \right ], K_{ij} = \tilde{K}_{ij} = \tilde{k}(b_{\color{Red} i}, b_{\color{Red} j}) \ $$ [[media: Fe1.s11.mtg16.djvu| (2) page 16-2 ]]

$$
 * <p style="text-align:right"> $$ \displaystyle
 * }

, where $$ {\color{Red} i} \ $$ and $$ {\color{Red} j} \ $$ are row  and  column, respectively.

[[media: Fe1.s11.mtg22.djvu| Page 22-4 ]]

Force/heat source
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$$ \tilde{\mathbf{F}}_{\color{Red} (n+1)x1} = \left \{ \frac{\mathbf{F}_E} {\mathbf{F}_F} \right \} \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(1)}
 * }

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$$ \mathbf{F}_{E_{\color{Red} 1x1}} = \left \{ F_{\color{Red} 0} \right \}, F_{\color{Red} 0} = \tilde{F}_{\color{Red} 0} = \tilde{f} (b_{\color{Red} 0}) \ $$, [[media: Fe1.s11.mtg16.djvu| (1) page 16-2 ]]

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(2)}
 * }

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$$ \mathbf{F}_{{\color{Red} F}_{\color{Red} nx1}} = \left \{ F_i, i = 1,...,n \right \}, F_i = \tilde{F}_ji = \tilde{f} (b_i) \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(3)}
 * }

 DWF - C [[media: Fe1.s11.mtg19.djvu|  page 19-3 ]] becomes (matrix form): 

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$$ \mathbf{b}_{\color{Red} (n+1)x1} (\beta) = \left \{ b_i (\beta) \right \} = \left \{ \frac{\mathbf{b}_E (\beta) } {\mathbf{b}_F (\beta) } \right \} = \left \{ \frac{\mathbf{b}_{\color{Red} 0} (\beta) } {\mathbf{b}_{i, \ i = 1,...,n} (\beta) }  \right \} =

\begin{Bmatrix} {\color{Red} 1} \\ - \\ {\color{Red} 0} \\ \vdots \\ {\color{Red} 0} \end{Bmatrix}

\ $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(7)}
 * }