User:Eml5526.s11.team5.JA/Mtg42

 Mtg 42: Fri, 15 Apr11

[[media: Fe1.s11.mtg42.djvu| Page 42-1 ]]

Constraint breaking solution:

Constraint breaking solution (CBS) - 1D: [[media: Fe1.s11.mtg20.djvu| (1) page 20-2 ]]


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[[media: Fe1.s11.mtg34.djvu| (1) and (3) page 34-3 ]] $$ w = 0 \ $$ on $$ \Gamma_g \Rightarrow \sum_{\color{Red}i} c_i b_i |_{\Gamma_g} \underbrace{=}_ 0 \ $$

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

NBF [[media: Fe1.s11.mtg26.djvu|   page 26-2 ]], [[media: Fe1.s11.mtg26.djvu| (1)  page 26-4 ]], [[media: Fe1.s11.mtg29.djvu| (1)-(2)  page 29-3 ]]

Select $$ b_{\color{Blue}i} = N_{\color{Blue}i} \ $$ such that


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$$ N_{\color{Blue}i}(\mathbf{x}_{\color{Blue}i}) = \delta_{\color{Blue}ij} \ $$

, where subscripts $$ {\color{Blue}i, j} \ $$ represent global node number

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


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$$ \Rightarrow N_{\color{Blue}i}(\mathbf{x}_{\color{Blue}i}) = 0, \ \forall \ \mathbf{x}_{\color{Blue}i} \in \Gamma_g, \ \forall \ \mathbf{x}_{\color{Blue}j} \notin \Gamma_g \ $$

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


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$$ \Rightarrow w^h(\mathbf{x}_{\color{Blue}j}) = \sum_{\color{Red}i} c_i \underbrace{ b_i (\mathbf{x}_{\color{Blue}j})}_{\color{Blue} \delta_{ij}} = c_{\color{Blue}j} \ $$

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


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$$ w^{\color{Red}h} = 0 \ $$ on $$ \Gamma_g \Leftrightarrow w^{\color{Red}h}(\mathbf{x}_{\color{Blue}i}) = 0, \ \forall \mathbf{x}_{\color{Blue}i} \in \Gamma_g \ $$

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


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$$ w^{\color{Red}h} = 0 \ $$ on $$ \Gamma_g \Leftrightarrow c_{\color{Blue}i} = 0, \ \forall \mathbf{x}_{\color{Blue}i} \in \Gamma_g \ $$

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

[[media: Fe1.s11.mtg42.djvu| Page 42-2 ]]

Similarly for $$ \tilde{\mathbf{d}} \ $$

Vibrating membrane: [[media: Fe1.s11.mtg38.djvu| page 38-4 ]] continued




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$$ \sum F_{\color{Red}x} = 0 = T(x + dx) cos(\theta(x+dx)) {\color{Red}-} T(x) cos(\theta(x)) \underbrace{\cong}_{{\color{Blue}\theta \ small} } (T+dT_{\color{Red}x}) \left ( 1 - \frac{d \theta^2}{2} \right ) - T \cdot {\color{Blue}1} \ $$

$$
 *  $$ \displaystyle
 * }


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$$ \Rightarrow dT_{\color{Red}x} = 0 \ $$

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

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$$ \sum F_{\color{Red}y} = 0 \Rightarrow dT_{\color{Red}y} = 0 \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle
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[[media: Fe1.s11.mtg42.djvu| Page 42-3 ]]

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$$ \Rightarrow T = {\color{Red}constant} \ $$

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

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$$ \underbrace{\sum F_{\color{Red}z}}_{\color{Blue} figure \ in \ page \ 38-4} = 0 = T {\color{Blue}dy} sin \theta_{\color{Red}x} (x+dx) {\color{Red}-} T {\color{Blue}dy} sin \theta_{\color{Red}x} (x) {\color{Red}+} T {\color{Blue}dx} sin \theta_{\color{Red}y} (y+dy) {\color{Red}-} T {\color{Blue}dx} sin \theta_{\color{Red}y} (y) {\color{Red}+} f {\color{Blue}dx \ dy} \ $$

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

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$$ \Rightarrow T \left ( \frac{\partial \theta_{\color{Red}x}}{\partial x} + \frac{\partial \theta_{\color{Red}y}}{\partial y} \right ) {\color{Red}+} f = 0 \ $$

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

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$$ \theta_{\color{Red}x} = \frac{\partial u}{\partial {\color{Red}x} }, \ \theta_{\color{Red}y} = \frac{\partial u}{\partial {\color{Red}y} } \ $$

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

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$$ T \underbrace{ \left ( \frac{\partial^2 u}{\partial {\color{Red}x}^2} + \frac{\partial^2 u}{\partial {\color{Red}y}^2} \right )}_{\color{Blue}{\rm div}(grad \ u )} {\color{Red}+} f = 0 \ $$

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

Dynamic: Instead of $$ f \ $$, use $$ f - \rho \frac{ \partial^{\color{Red}2} u } { \partial t^{\color{Red}2} } \Rightarrow \ $$ [[media: Fe1.s11.mtg40.djvu| (1)  page 40-4 ]]

[[media: Fe1.s11.mtg42.djvu| Page 42-4 ]]

Alternative derivation:



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$$ \sum F_{\color{Red}z} = 0 = \int_{\partial \omega} \mathbf{n} \cdot grad(u) \ d(\partial \omega) + \int_{\omega} f \ d \omega = 0 \ \forall \omega \in \Omega $$

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

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$$ \sum F_{\color{Red}z} = T \int_{\omega}{\rm div}(grad \ u) \ d(\partial \omega) + \int_{\omega} f \ d \omega = 0 \ \forall \omega \in \Omega $$

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

$$ \Rightarrow \ $$ [[media: Fe1.s11.mtg42.djvu| (2) page 42-3 ]]