User:Eml5526.s11.team5.JA/Mtg10

 Mtg 10: Sat, 22 Jan 11

[[media: Fe1.s11.mtg10.djvu| Page 10-1 ]]

WRF  continued [[media: Fe1.s11.mtg8.djvu|   page 8-2 ]]

Family of linearly independent basis functions $$ {\color{Blue} \left \{ b_1(x), b_2(x),...., b_n(x), ...., b_{\color{Red} \infty}(x)  \right \} = F } \ $$

example, Fourier basis functions $$ \left \{ 1, cos(iwx),sin(iwx) \right \}  \ $$


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$$ f(x) \ $$ periodic $$ \Rightarrow f(x+ \underbrace{T}_{\color{Blue} period} ) = f(x) \ $$

$$
 *  $$ \displaystyle
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$$ f(x)= \alpha_0 + \sum_{i=1}^{\infty}\alpha_i.cos(iwx) + \sum_{i=1}^{\infty}\beta_i.sin(iwx) \ $$

$$
 *  $$ \displaystyle
 * }

[[media: Fe1.s11.mtg10.djvu| Page 10-2 ]]


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$$ 2 \pi = \omega T \Rightarrow \omega = \frac{2 \pi}{T} \ $$

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

Another example: $$ \underbrace{ \left \{ 1, x,x^2, x^3,.... \right \} }_{\color{Blue} \left \{ x^0, x^1,x^2, x^3,.... \right \} }\ $$

Test of linear independence of F: Gram matrix


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$$ \boldsymbol{\Gamma}( \underbrace{b_1(x), ..., b_n(x)}_{\color{Blue}F}) = \left [ \underbrace{  }_{\color{Blue} \Gamma_{ij}} \right ]_{\color{Red}nxn} \ $$

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

, where $$  = {\color{Blue}\Gamma_{ij}} \ $$ was defined on [[media: Fe1.s11.mtg8.djvu| (5) page 8-3 ]]


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$$ \Gamma_{ij} =  = \int_{\Omega} b_i(x) b_j(x) \ dx \ $$

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


 * }

[[media: Fe1.s11.mtg10.djvu| Page 10-3 ]]

1) Construct $$ \boldsymbol{\Gamma}_{\color{Red}5x5}(F) \ $$; observe properties of $$ \boldsymbol{\Gamma} \ $$ (diagonal)

2) Find $$ det \ \boldsymbol{\Gamma}(F) \ $$

3) Conclude $$ F \ $$ is  orthogonal  basis, example,


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$$ \Gamma_{ij} =  = \delta_{ij} \ $$ [[media: Fe1.s11.mtg7.djvu| (2) page 7-1 ]]

$$
 * <p style="text-align:right"> $$ \displaystyle
 * }
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Continuous WRF  [[media: Fe1.s11.mtg6.djvu| (4)  page 6-4 ]]

Discrete WRF

$$ F = \left \{ b_i(x), i = 1,....,n \right \} \ $$, family of linearly independent basis functions

Approximately $$ w(x) \ $$ and $$ u(x) \ $$ by: (F&B p.84)

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$$ w(x) \cong w^h(x) = \sum_{i=1}^{n} c_ib_i(x) \ $$

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

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$$ u(x) \cong u^h(x) = \sum_{i=1}^{n} d_jb_j(x) \ $$

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

[[media: Fe1.s11.mtg10.djvu| Page 10-4 ]]

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(4) p. 6-4 $$ \Rightarrow \int_{\Omega} w^h(x)P(u^h(x)) dx = 0  \forall w^h(x) \ $$

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

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(1) p. 10-3 $$ \Rightarrow \int_{\Omega}b_i(x) P(u^h(x)) dx = 0, i = 1,...n  \ $$

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

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$$ P(u) \ $$ is linear in $$ u \ $$ ( [[media: Fe1.s11.mtg6.djvu| (2) page 6-4 ]], [[media: Fe1.s11.mtg9.djvu| (4)  page 9-1 ]] ) $$
 * <p style="text-align:right"> $$ \displaystyle
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$$ P(u^h) = P(\sum_{j=1}^{n} d_jb_j(x)) = \sum_{j=1}^{n} d_j P( b_j(x)) \ $$

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

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(2) & (3) $$ \Rightarrow \sum_{j=1}^{n} \left (\int_{\Omega} b_i P( b_j(x)) dx \right ) d_j = 0, i = 1,....,n \ $$

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

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$ {\color{Blue}n} \ $ equations for $ {\color{Blue}n} \ $ unknows $ {\color{Blue} \left \{ d_j \right \}_{\color{Red}nx1} } \ $

$$
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[[media: Fe1.s11.mtg6.djvu| (2) page 6-4 ]] & [[media: Fe1.s11.mtg9.djvu| (4)  page 9-1 ]]: $$ P(u) \ $$ for static

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$$ P(u) = \sum_{j=1}^{n} \left ( \int_{\Omega} b_i \left [ \frac{d}{dx} \left ( a_2 \frac{d}{dx} b_j(x) \right ) \right ] dx \right ) d_j \ $$

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

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$$ P(u) = - \int_{\Omega} b_i(x) f(x) dx  {\color{Blue} =:F_i} \ $$

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

[[media: Fe1.s11.mtg10.djvu| Page 10-5 ]]

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$$ \Rightarrow \sum_{j=1}^{n} \bar{K}_{ij}d_j = F_i, i = 1,...,n \ $$

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

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$$ \bar{K}_{ij} := \int_{\Omega} b_i \left [ \frac{d}{dx} \left ( a_2 \frac{d}{dx} b_j(x) \right ) \right ] dx \ $$

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

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$$ \Rightarrow \left [ \bar{K}_{ij} \right ]_{\color{Red}nxn} \left \{ d_{j} \right \}_{\color{Red}nx1} = \left \{ f_{j}  \right \}_{\color{Red}nx1} \ $$

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

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$$ \Rightarrow \bar{\mathbf{K}}_{\color{Red}nxn} \mathbf{d}_{\color{Red}nx1} = \mathbf{F}_{\color{Red}nx1} \ $$

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