User:Eml5526.s11.team5.JA/Mtg20

Mtg 20: Wed, 16 Feb 11

[[media: Fe1.s11.mtg20.djvu| Page 20-1 ]]

Problem: $$ \left \{ c_i  \right \} \ $$ in [[media: Fe1.s11.mtg19.djvu| (2)  page 19-3 ]]  (DWF)   are no longer "free" due to constraint [[media: Fe1.s11.mtg19.djvu| (3)  page 19-3 ]], example, $$ \left \{ c_i  \right \} \ $$  cannot  be selected arbitrarily as done with


 * $$ \left \{ \alpha_i \right \} \ $$ in the vector case [[media: Fe1.s11.mtg8.djvu|   page 8-3 ]], and


 * $$ \left \{ c_i \right \} \ $$ in the DWRF case [[media: Fe1.s11.mtg10.djvu|   page 10-3 ]], [[media: Fe1.s11.mtg10.djvu|   page 10-4 ]]

Consider $$ \left \{ c_i, i = {\color{Red}0}, 1,...,n \right \}  = \underbrace{ \left \{ 1, 0,...,0  \right \} }_{\color{Blue }\left \{ c_0, c_1,...,c_n  \right \}} \ $$

[[media: Fe1.s11.mtg19.djvu| (3) page 19-3 ]]: $$ w^h(\beta) = \sum_{i={\color{Red}0}}{n} c_i b_i (\beta) = c_0 b_0 (\beta) {\color{Red} \ne \mathbf{0} }  \ $$

in general since $$ b_0(.) \ $$ is typically a constant, example:


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$$ b_0(x) = 1 \ \forall \ x \ \Rightarrow b_0(\beta) = 1 {\color{Red} \ne \mathbf{0} } \Rightarrow w^h(\beta){\color{Red} \ne \mathbf{0} } \ $$  contrary  to [[media: Fe1.s11.mtg19.djvu| (5) page 19-2 ]]

$$
 *  $$ \displaystyle
 * }

[[media: Fe1.s11.mtg20.djvu| Page 20-2 ]]

Constraint Breaking Solution (CBS)
Choose appropriate basis functions to "destroy' the constraint [[media: Fe1.s11.mtg19.djvu| (3) page 19-3 ]] example:


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$$ \left \{ b_i(\beta), i = {\color{Red}0}, 1,...,n \right \}  = \left \{ \underbrace{b_{\color{Red}0}(\beta)}_{\color{Blue} \ne 0}, \underbrace{b_1(\beta)}_{\color{Red} = 0},...,\underbrace{b_n(\beta)}_{\color{Red} = 0}  \right \} =  \left \{ b_{\color{Red}0}(\beta), 0,...,0  \right \} \ $$

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

[[media: Fe1.s11.mtg19.djvu| (3) page 19-3 ]]: $$ w^h(\beta) = c_{\color{Red}0} b_{\color{Red}0}(\beta) {\color{Red} = 0} \ $$

[[media: Fe1.s11.mtg19.djvu| (3) page 19-3 ]], $$ w^h(\beta) = \sum_{i={\color{Red}0}}^n c_i b_i (\beta) = \underbrace{c_{\color{Red}0}}_{\color{Red} = 0}  \underbrace{b_{\color{Red}0} (\beta)}_{\color{Red} \ne 0}  + c_1 \underbrace{b_1 (\beta)}_{\color{Red} = 0}  + ..... + c_n \underbrace{b_n (\beta)}_{\color{Red} = 0}    {\color{Red} = 0} \ $$   for any choice of $$  {\color{Blue} \left \{ c_1, c_2, ... , c_n \right \} } \ $$

[[media: Fe1.s11.mtg20.djvu| Page 20-3 ]]

Note:  [[media: Fe1.s11.mtg20.djvu| (3)  page 20-2 ]]  can be written as


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$$ \mathbf{c}_{\color{Red}E} = \left \{ c_{\color{Red}0} \right \} = \left \{ 0  \right \} \ $$ homogeneous essential boundary condition

$$
 *  $$ \displaystyle
 * }

, where subscript $$ {\color{Red}E} \ $$ stand for "Essencial"; see F&B p. 21

Application: Choice of basis functions in F&B p. 72 pb. 3.3 & pb. 3.4:  $$ \Gamma_g = \left \{ \beta  \right \} = \left \{ x = 3  \right \}, g = 0.001 \ $$


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$$ \left \{ b_i(x), i = {\color{Red}0}, 1,...,n \right \} = \left \{ (x-3)^{\color{Red}i}, i = {\color{Red}0}, 1,...,n  \right \} = \left \{ {\color{Red}1}, (x-3), (x-3)^2,..., (x-3)^n \right \} \ $$

$$
 *  $$ \displaystyle
 * }


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$$ b_{\color{Red}0}(3) = 1 \ne 0 \ $$

$$
 *  $$ \displaystyle
 * }


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$$ b_i(3) = (3-3)^i = 0, i = 1,2,..,n \ $$

$$
 *  $$ \displaystyle
 * }