User:Eml5526.s11.team5.JA/Mtg30

 Mtg 30: Mon, 14 Mar 11


 * Added extra comment at 30-2 page beginning

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 Problem:  LIBF as   GBS   still cannot address problem of  complex geometry    (rectangle, [[media: Fe1.s11.mtg29.djvu|     page 29-2 ]]) 

 Solution:   Subdivide $$ \Omega \ $$ ([[media: Fe1.s11.mtg25.djvu|     page 25-1 ]], [[media: Fe1.s11.mtg23.djvu|     page 23-1 ]]) (disk) into sub-domains of arbitrary shape -   elements . Example: quadrilaterals



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This method can be applied to multi-element meshes



note that function $$ \varphi\ $$ is different for each mesh element, since it adjusts same dimensioned parent element to different shaped child (meshed) elements.


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$$ w^{(1)} = \varphi^{(1)}(\bar{w}), \ etc. \ $$

$$
 *  $$ \displaystyle
 * }

How to construct $$ \varphi^{(e)} \ $$? ''' Use LIBF! '''

 1-D: Quadratic elements 



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$$ x^{\color{Blue}e} \ $$ coord. $$ x \ $$ in $$ w^{\color{Blue}e} \ $$


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$$ In \ {\color{Blue}w^{(1)}} : \ x^{(1)} = \varphi^{(1)}(\xi) = \sum_{i = 1}^3 L_{i,3}(\xi) x_i \ $$

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


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$$ In \ {\color{Blue}w^{(2)}}: \ x^{\color{Red}(2)} = \varphi^{\color{Red}(2)}(\xi) = \sum_{i = 1}^3 L_{i,3}(\xi) x_{\color{Red}i+2} \ $$

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


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$$ In \ {\color{Blue}w^{(3)}}: \ x^{\color{Red}(3)} = \varphi^{\color{Red}(3)}(\xi) = \sum_{i = 1}^3 L_{i,3}(\xi) x_{\color{Red}i+4} \ $$

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

 In general 

, where


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$$ {\color{Red}I} = {\color{Red}Global} \ node \ number \ $$

$$
 *  $$ \displaystyle
 * }


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$$ {\color{Blue}i} = {\color{Blue}Local } \ node \ no. \ $$

$$
 *  $$ \displaystyle
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$$ {\color{Red}3} = nnel \ (number \ of \ \ nodes \ per \ elem) \ $$

$$
 *  $$ \displaystyle
 * }

Also:

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 Notation 


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$$ x^e \ $$ : superscript "e" designates  element "e" 

$$
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$$ x^{{\color{Red}(}2{\color{Red})}} \ $$: superscript "{\color{Red}(}2{\color{Red})}" (number in parentheses) designates "element 2"

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

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$$ x^{\color{Red}2} \ $$: superscript "2" (number without parentheses) designates order number of coordinates, so $$ x^{\color{Red}2} \equiv y \ (x^{\color{Red}1} \equiv x, x^{\color{Red}3} \equiv z) \ $$

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

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For  polynomial , $$ x^{\color{Red}2} \equiv \ ' x \ squared' \ $$

$$
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$$ {\color{Blue}( } x^{\color{Red}2}  {\color{Blue} ) }^{\color{Blue}2} \equiv y^{\color{Red}2} \ ('y \ squared') \ $$

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

 Global matrices

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$$ \Omega = \overset{nel}{\underset{e=1}{\cup}} w^e \ $$

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

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$$ K_{ij} = \int_{\Omega} b_i'(x) a_2(x) b_j'(x) dx = \sum_{e=1}^{nel} \underbrace{ \int_{\omega^e}b_i^{e'}(x^e) a_2(x^e) b_j^{e'}(x^e) dx^e}_{\color{Blue} K_{ij}^e} \ $$

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

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$$ K_{ij}^{\color{Blue}e} = b_i^(x^{\color{Blue}e}) a_2(x^{\color{Blue}e}) b_j^(x^{\color{Blue}e}) dx \ $$

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

 Anatomy of notation 

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$$ K_{ij}^{\color{Blue}e} \ and \ b_i^ \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle
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$$ K \ $$ = "big K" = global

$$
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$$ ij \ $$ and $$ i \ $$ = global degree of freedom number

$$
 * <p style="text-align:right"> $$ \displaystyle
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$$ {\color{Blue}e} \ $$ = element e

$$
 * <p style="text-align:right"> $$ \displaystyle
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$$ ^{\color{Red}'} \ $$ = derivative

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

Example: Linear Lagrangian element basis functions (  L  L  E  BF)



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