User:Eml5526.s11.team5.JA/Mtg14

Mtg 14: Mon, 31 Jan 11

[[media: Fe1.s11.mtg14.djvu| Page 14-1 ]]

- Team Questions

- mediawiki category: Demo

- Read Chapter 2, Section 2.1-2.4, Examples 2.3

Overview of EML 4500 F08 FEA D

Pros & Cons of Weighted Residue Form (WRF)
Pros: Direct generalization from vectors (computing of components) to functions

WRF = enforcing orthogonality of residue $$ P(u^h) \ $$ with respect to (wrt) all basis functions in $$ \left \{ b_j(x) \right \} \ $$

Conceptual interpretation

[[media: Fe1.s11.mtg12.djvu| (4) page 12-2 ]], [[media: Fe1.s11.mtg10.djvu| (2)  page 10-4 ]]

[[media: Fe1.s11.mtg14.djvu| Page 14-2 ]]

Cons:

$$ \mathbf{A)} \ $$ [[media: Fe1.s11.mtg10.djvu| (2) page 10-5 ]]: $$ \bar{\mathbf{K}}^{\color{Red}T} \ne \bar{\mathbf{K}} \ $$ (not symmetric)


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$$ \bar{\mathbf{K}} = [\bar{K}_] \ $$, where $$ {\color{blue}i = row } \ $$, $$ {\color{Red}j = column} \ $$

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$$ \bar{K}_ \ne \bar{K}_ \ $$ [[media: Fe1.s11.mtg10.djvu| (2) page 10-5 ]]

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$$ \bar{K}_ := \int_{\Omega} b_{\color{Red}j} (x) \underbrace{ \left [ \frac{d}{dx} \left ( a_2 \frac{d}{dx} b_{\color{blue}i}(x) \right ) \right ]}_{ \underbrace{ \left ( a_2 b_i {\color{Red}'} \right ){\color{Red}'} }_{a_2'{\color{Red}'} b_i{\color{Red}'} + a_2 b_i{\color{Red}''}}   } dx \ $$

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$$ {\color{Blue}\mathbb{A}} := b_i [ a_2' b_j' + a_2 b_j'' ] \ $$

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$$ {\color{Blue} \mathbb{B}} := b_j [ a_2' b_i' + a_2 b_i'' ] \ $$

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[[media: Fe1.s11.mtg14.djvu| Page 14-3 ]]

$$ \mathbf{B)} \ u^h $$ (thus $$ b_j, j = 1,...,n \ $$ ) must be at least $$ \underbrace{twice}_{ {\color{Red} stronger \ requirement \ \Rightarrow} {\color{Blue} \ more \ conditions \ (2)} }   $$ differentiable

$$ \mathbf{C)} \ u^h $$  must satisfy not just essential boundary condition, but also the natural boundary condition (1-D)

$$ \mathbf{D)} \  $$ Difficult (impossible) to use for problems with complex geometry