User:Eml5526.s11.team5.JA/Mtg16

Mtg 16: Mon, 7 Feb 11

[[media: Fe1.s11.mtg16.djvu| Page 16-1 ]]


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

[[media: Fe1.s11.mtg15.djvu| (2) page 15-3 ]]: $$ {\color{Blue}\mathbb{K}} = \underbrace{\color{Blue}\mathbb{K}\mathbf{1}}_{\color{Blue} boundary \ term} - \underbrace{\color{Blue}\mathbb{K}\mathbf{2}}_{\color{Blue} internal \ term} \ $$

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


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$${\color{Blue}\mathbb{K}\mathbf{1}} = \left ( w \underbrace{a_2 \frac{\partial u}{\partial x}}_{\color{Red}unknown} \right )_{\color{Blue}x = \beta} {\color{Red}-} \left ( w \underbrace{a_2 \frac{\partial u}{\partial x}}_{\color{Red} -h = known} \right )_{\color{Blue}x = \alpha} \ $$

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

IMPORTANT:  Since $$ w \ $$ is  arbitrary , select $$ w \ $$ such that:

to  conveniently  remove the  unknown  "flux" at $$ x = \beta \ $$, example $$ u(\beta) = g \ $$ ([[media: Fe1.s11.mtg15.djvu| (1)  page 15-2 ]])

Putting together:  (1)-(3) & [[media: Fe1.s11.mtg15.djvu| (2)  page 15-3 ]]


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ \Rightarrow WRF = {\color{Blue}\mathbb{K}\mathbf{1}} - {\color{Blue}\mathbb{K}\mathbf{2}} + \mathbb{F} - \mathbb{M} = w(\alpha)h - \underbrace{\int_{\alpha}^{\beta} \frac{\partial w}{\partial x} a_2 \frac{\partial u}{\partial x} dx}_{\color{Blue}\tilde{k}(w,u)} + \int_{\alpha}^{\beta} wf dx - \underbrace{\int_{\alpha}^{\beta} w \bar{m} u^{\color{Red}(s)} dx}_{\color{Blue}\tilde{m}(w,u^)} \ $$

$$
 *  $$ \displaystyle {\color{Red}(0)}
 * }

,note 
 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ u^{\color{Red}(s)} = \frac{\partial^{\color{Red}s} u}{\partial t^{\color{Red}s}} \ $$

$$
 *  $$ \displaystyle
 * }

[[media: Fe1.s11.mtg16.djvu| Page 16-2 ]]

Define:


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ {\color{Blue}\tilde{f}(w) := } w(\alpha)h + \int_{\alpha}^{\beta} wf dx \ $$

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


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ {\color{Blue}\tilde{k}(w,u) := } \int_{\alpha}^{\beta} \frac{\partial w}{\partial x} a_2 \frac{\partial u}{\partial x} dx \ $$

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


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ {\color{Blue}\tilde{m}(w,u^{\color{Red}(s)}) := } \int_{\alpha}^{\beta} w \bar{m} u^{\color{Red}(s)} dx \ $$

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

(continued) Weighted Form (WF) (c.f. F&B, p.49, (3.20))

$$ {\color{Blue}\tilde{m}(w,u^{\color{Red}(s)})} \ $$ = mass/capacitance operator linear in $$ (w,u^{\color{Red}(s)}) \ $$

$$ {\color{Blue}\tilde{k}(w,u)} \ $$ = stiffness/conductivityoperator linear in $$ (w,u) \ $$

$$ {\color{Blue}\tilde{f}(w)} \ $$ = force/heat source operator linear in $$ w \ $$

[[media: Fe1.s11.mtg16.djvu| Page 16-3 ]]

Note:  Consider $$ \tilde{k}(w,u) \ $$ [[media: Fe1.s11.mtg16.djvu| (2)  page 16-2 ]]

1) both $$ w \ $$ and $$ u \ $$ must be differential once.

$$ \Rightarrow \ $$ Differentiability (or smoothness) requirement on $$ u \ $$ is

-  weaker  (less) in WF [[media: Fe1.s11.mtg16.djvu| (4) page 16-2 ]]

-  stronger  (more) in PDE [[media: Fe1.s11.mtg4.djvu| (4) page 4-4 ]]

-  stronger  (more) in WRF [[media: Fe1.s11.mtg6.djvu| (4) page 6-4 ]]

2) Natural boundary condition [[media: Fe1.s11.mtg4.djvu| (4)  page 4-4 ]]  absorbed  into WF;

$$ u \ $$ ( and thus $$ u^h \ $$ )  not  required to satisfy natural boundary condition (unlike solving PDE or WRF), but required to satisfy  only  essential boundary condition

[[media: Fe1.s11.mtg16.djvu| Page 16-4 ]]

3) $$ w \ $$ ( and thus $$ w^h \ $$ required to satisfy homogeneous essential boundary condition