User:Eml5526.s11.team5.JA/Mtg26

Mtg 26: Wed, 2 Mar 11

[[media: Fe1.s11.mtg26.djvu| Page 26-1 ]]

Dynamic reaction (general flux at essential boundary condition $ {\color{Blue}\Gamma_g } \ $): continued


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$$ \mathbf{P}_E := \left [ \left ( \mathbf{M}_{E{\color{Blue}E}} \cdot {\color{Red}g}^ + \mathbf{K}_{E{\color{Blue}E}} \cdot {\color{Red}g} \right ) + \mathbf{M}_{E{\color{Blue}F}} \mathbf{d}_{\color{Blue}F}^{\color{Red}(s)} + \mathbf{K}_{E{\color{Blue}F}} \mathbf{d}_{\color{Blue}F} - \mathbf{F}_{E} \right ] = \mathbf{r}_{E} \ $$,  [[media: Fe1.s11.mtg25.djvu| (1) & (2)  page 25-1 ]]

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

[[media: Fe1.s11.mtg16.djvu| (2) page 16-1 ]], [[media: Fe1.s11.mtg16.djvu| (0)  page 16-2 ]]:


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$$ {\color{Blue}+} \left ( w \cdot \underbrace_{\color{Red}outward \ normal} a_2 \frac{\partial u}{\partial x} \right )_{\color{Blue}x = \beta} {\color{Red} = } \ \tilde{m}(w,u^{\color{Red}(s)}) + \tilde{k}(w,u) - \tilde{f}(w) \ $$

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


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$$ \mathbf{P}_E := \left [ \underbrace{ \mathbf{M}_{E{\color{Blue}E}} }_{\color{Blue}known \ before \ \mathbf{d}_F \ is \ solved \ for } \cdot {\color{Red}g}^ + \mathbf{M}_{E{\color{Blue}F}} \mathbf{d}_{\color{Blue}F}^  \right ] + \left [ \underbrace{\mathbf{K}_{E{\color{Blue}E}}}_{\color{Blue}known \ before \ \mathbf{d}_F \ is \ solved \ for } \cdot {\color{Red}g}  +  \mathbf{K}_{E{\color{Blue}F}} \mathbf{d}_{\color{Blue}F} \right ] - \mathbf{F}_{E}  \ $$

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

,where


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$$ \tilde{m}(w,u^{(s)}) = \left [ \mathbf{M}_{ E{\color{Blue}E}} \cdot {\color{Red}g}^ +  \mathbf{M}_{E{\color{Blue}F}} \mathbf{d}_{\color{Blue}F}^  \right ] \ $$

$$
 *  $$ \displaystyle
 * }


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$$ \tilde{k}(w,u) = \left [ \mathbf{K}_{E{\color{Blue}E}} \cdot {\color{Red}g} +  \mathbf{K}_{E{\color{Blue}F}} \mathbf{d}_{\color{Blue}F} \right ]  \ $$

$$
 *  $$ \displaystyle
 * }


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$$ \tilde{f}(w) = \mathbf{F}_{E} \ $$

$$
 *  $$ \displaystyle
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Static


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$$ \mathbf{P}_E := \left [ \mathbf{K}_{E{\color{Blue}E}} \cdot {\color{Red}g}  +  \mathbf{K}_{E{\color{Blue}F}} \mathbf{d}_{\color{Blue}F} \right ] - \mathbf{F}_{E} =  \mathbf{r}_{E} \  $$   F&B, p.107 (5.34)

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

(end dynamic reaction, post procedure for generated flux @ $$ \Gamma_g \ $$ )

[[media: Fe1.s11.mtg26.djvu| Page 26-2 ]]

Nodal Basis Funcs (NBS) [[media: Fe1.s11.mtg25.djvu| page 25-1 ]] continued

1) Lagrange interpolation basis function (LIBF)   $$ L_{i,n}(x) \Rightarrow \ $$ 1 degree of freedom (dof) per node

2) Hermite interpolation basis function (HIBF)  $$ H_{i,2}(x) \Rightarrow \ $$ several degrees of freedom (dofs) per node

G1DM1.0/D1:  [[media: Fe1.s11.mtg9.djvu| (3)  page 9-2 ]] $$ \Gamma_g = \{ 1 \}, \Gamma_h = \{ 0 \} \ $$

G1DM1.0/D1b:  Same as  G1DM1.0/D1, except  $$ \Gamma_g = \{ {\color{Red}0} \}, \Gamma_h = \{ {\color{Red}1} \} \ $$,  HW 4.4 p. 21-1

Note: Connection between $$ F_F \ $$ and $$ F_e \ $$


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$$ e^{n \Theta} = cos(n \Theta) + {\color{Blue}i} \cdot sin(n \Theta) \ $$, where $$ {\color{Blue}i = \sqrt{-1}} \ $$  (Euler, de Moivre)

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

[[media: Fe1.s11.mtg26.djvu| Page 26-3 ]]



Example:

<span id="(1)">
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$$ n = 4, \ x_1 = 1, \ x_2 = 2, \ x_3 = 3, \ x_4 = 4 \ $$

$$
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<span id="(1)">
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$$ L_{{\color{Blue}i},{\color{Red}m}} = L_{{\color{Blue}3},{\color{Red}4}}(x) := \frac{(x-x_1)(x-x_2)(x-x_4)}{(x_{\color{Red}3}-x_1)(x_{\color{Red}3}-x_2)(x_{\color{Red}3}-x_4)} = \frac{(x-{\color{Blue}1})(x-{\color{Blue}2})(x-{\color{Blue}4})}{({\color{Red}3}-1)({\color{Red}3}-2)({\color{Red}3}-4)}\ $$

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

[[media: Fe1.s11.mtg26.djvu| Page 26-4 ]]



Note: $$ L_{3,4}(x_{max}) {\color{Red}\ne} 1 \ $$, actually it's  greater  then $$ 1 \ $$ and $$ x_{max} \ne x_3 \ $$ but (!) at $$ x_3, L_{3,4}(x_3) = 1 \ $$

Also:

, where $$ \delta \ is {\color{Blue}\ Kronecker \ delta } \Rightarrow

\delta_{3,j} = \begin{cases} 1 & for j = 3 \\ 0, & for j \ne 3 \end{cases}

\ $$

End Example