User:Eml5526.s11.team01.roark/Mtg39

[[media: Fe1.s11.mtg39.djvu| Mtg 39]]: Fri, 8 Apr 11 [[media: Fe1.s11.mtg39.djvu| Page 39-1]]
 * Comments on HW6 (see [[media: Fe1.s11.mtg39.djvu| p. 39-6]]

HW 7.1:

(HW6.5 Continued, 2D LIBF) Initial Condition
 * 1) Static (steady state): $$\displaystyle f\left( x \right)=1$$ in $$\displaystyle \Omega =\square $$
 * 2) Dynamic (Transient): $$\displaystyle \rho c=3$$
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$$\displaystyle u\left(\mathbf x,t=0 \right)=xy\,\,\forall \mathbf x\in \Omega $$
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2a. $$\displaystyle \begin{matrix} f\left(\mathbf x,t \right)=0,\,\forall \mathbf x\in \Omega ,\,\forall t>0 \\ g\left(\mathbf x,t \right)=2\,on\,{{\Gamma }_{g}}=\partial \Omega  \\ \end{matrix}$$

2b. $$\displaystyle \begin{matrix} f\left(\mathbf x,t \right)=1,\,\forall \mathbf x\in \Omega ,\,\forall t>0 \\ g\left(\mathbf x,t \right)=2\,on\,{{\Gamma }_{g}}=\partial \Omega  \\ \end{matrix}$$

Plot $$\displaystyle {{u}^{h}}$$ at center vs. t till convergence to steady state.

End HW 7.1

Transient analysis (heat)

[[media: Fe1.s11.mtg23.djvu|(3) p. 23-3]], s=1


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$$\displaystyle \mathbf M\dot{\mathbf d}+\mathbf K \mathbf d=F\Rightarrow \dot{\mathbf d}={{\mathbf M}^{-1}}\left[ \mathbf F-\mathbf K\mathbf d \right]$$ [[media: Fe1.s11.mtg39.djvu| Page 39-2]] Initial condition
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$$\displaystyle u\left( \mathbf x,t=0 \right)={{u}_{0}}\left( x \right)\,\,\forall \mathbf x\in \Omega $$ where $$\displaystyle {{u}_{0}}\left( x \right)$$ e. g., [[media: Fe1.s11.mtg39.djvu|(1) p. 39-1]]
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Projection (1) on $$\displaystyle ~\left\{ {{b}_{i}} \right\}$$: Recall [[media: Fe1.s11.mtg15.djvu| p. 15-1]] Summary, [[media: Fe1.s11.mtg10.djvu|(1) p. 10-3]] $$\displaystyle {{w}^{h}}$$ and [[media: Fe1.s11.mtg10.djvu|(2) p. 10-3]] $$\displaystyle {{u}^{h}}$$

(1):
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$$\displaystyle \left\langle w,u\left(\mathbf x,t=0 \right) \right\rangle =\left\langle w,{{u}_{0}}\left(\mathbf x \right) \right\rangle \,\,\forall w$$ Note: $$\displaystyle \forall \mathbf x\in \Omega $$, i.e., $$\displaystyle \mathbf x\notin \partial \Omega $$
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(2) and [[media: Fe1.s11.mtg10.djvu|(1)-(2) p. 10-3]]:
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Find $$\displaystyle \left\{ {{d}_{j}}\left( 0 \right)\,\,\forall {{\mathbf x}_{j}}\notin \Omega \right\}={{\mathbf d}_{F}}\left( 0 \right)$$ st
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$$\displaystyle \sum\limits_{i}\left[ \sum\limits_{j}{\underbrace{\left\langle {{b}_{i}},{{b}_{j}} \right\rangle }_{{d}_{j}}\left( 0 \right)-\underbrace{\left\langle {{b}_{i}},{{u}_{0}} \right\rangle }_} \right]=0$$
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where $$\displaystyle {{\Gamma }_{ij}}$$ is from [[media: Fe1.s11.mtg10.djvu|(3) p. 10-2]]

$$\displaystyle \forall \left\{ {{c}_{i}}\forall {{\mathbf x}_{i}}\in \Omega \right\}={{\mathbf c}_{F}}$$
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[[media: Fe1.s11.mtg39.djvu| Page 39-3]] Recall
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$$\displaystyle {{\mathbf d}_{F}}\left( t \right)= \mathbf d\left( t \right)$$ (see [[media: Fe1.s11.mtg23.djvu|(2) p. 23-3]])
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[[media: Fe1.s11.mtg39.djvu|(4) p. 39-2]] : Find $$\displaystyle {{\mathbf d}_{F}}\left( 0 \right)$$ st
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$$\displaystyle {{\mathbf c}_{F}}\cdot \left[ {{\mathbf \Gamma }_{FF}}{{\mathbf d}_{F}}\left( 0 \right)- \mathbf G \right]=0\,\,\forall {{\mathbf c}_{F}}$$
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$$\displaystyle \mathbf d\left( 0 \right)={{ \mathbf d}_{F}}\left( 0 \right)= \mathbf \Gamma _{FF}^{-1}\mathbf G$$
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$$\displaystyle \eta $$ = set of all global node numbers

$$\displaystyle {{\eta }_{g}}$$ = set of global node numbers on $$\displaystyle {{\Gamma }_{g}}$$

$$\displaystyle {{\eta }_{h}}$$ = set of global node numbers on $$\displaystyle {{\Gamma }_{h}}$$

$$\displaystyle {{\eta }_{H}}$$ = set of global node numbers on $$\displaystyle {{\Gamma }_{H}}$$
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$$\displaystyle {{\eta }_{F}}$$ = $$\displaystyle \eta $$ \  $$\displaystyle \left( {{\eta }_{g}} \right.$$ Strikethrough: $$\displaystyle \left. \cup {{\eta }_{h}}\cup {{\eta }_{H}} \right)$$ $$\displaystyle {{\eta }_{F}}$$ = set of global node numbers NOT on $$\displaystyle {{\Gamma }_{g}}$$ or on $$\displaystyle {{\Gamma }_{h}}$$ or on $$\displaystyle {{\Gamma }_{H}}$$ (Free) [[media: Fe1.s11.mtg39.djvu| Page 39-4]]
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Example 1:



Boundary conditions [[media: Fe1.s11.mtg38.djvu| p. 38-2]]

Mesh [[media: Fe1.s11.mtg29.djvu| p. 29-2]]

$$\displaystyle \eta =\left\{ 1,2,3,...,16 \right\}$$



$$\displaystyle {{\eta }_{F}}=\left\{ 5,6,7,9,10,11,13,14,15 \right\}$$

$$\displaystyle \underbrace_{9\times 9}=\left[ \left\langle b\underbrace{_{i}}_{Row},b\underbrace{_{j}}_{Col} \right\rangle ;\,\,i,j\in {{\eta }_{F}} \right]$$

End Example

[[media: Fe1.s11.mtg39.djvu| Page 39-5]]