User:Eml5526.s11.team5.JA/Mtg34

Mtg 34: Wed, 23 Mar 11

[[media: Fe1.s11.mtg34.djvu| Page 34-1 ]]

2D, 3D scalar field problems: continued [[media: Fe1.s11.mtg33.djvu| page 33-2 ]]


Boundary Conditions:


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$$ \partial \Omega = \underbrace{\color{Green}\Gamma_g}_{\color{Blue} essential} \cup \underbrace{\color{Red}\Gamma_h}_{\color{Blue} natural} \ $$

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

G2DM1.0:

PDE [[media: Fe1.s11.mtg33.djvu| (4) page 33-2 ]]

Essential boundary condition (2)

Natural boundary condition (3)


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[[media: Fe1.s11.mtg34.djvu| Page 34-2 ]]

Note: G2DM1.0 reduced  exactly  to  G1DM1.0.

WRF:


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$$ \int_{\Omega} {\color{Red}w} \underbrace{ \left [ {\rm div}( \boldsymbol{\kappa} \cdot grad(u)) + f {\color{Red}-} \rho c \frac{ \partial u}{\partial t} \right ] }_{\color{Blue}residual \ P(u) \equiv r(u) \ in \ FB} d \Omega = 0 \ $$
 *  $$ \displaystyle {\color{Red}(1)} $$


 * }

Integrating by parts:


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$$ {\color{Blue} \mathbb{K}} := \int_{\Omega} w \underbrace{{\rm div}( \boldsymbol{\kappa} \cdot grad(u)) }_{\color{Blue}\frac{ \partial }{\partial x_i} \left ( \kappa_{ij} \frac{ \partial u }{\partial x_j} \right ) } d \Omega \ $$
 *  $$ \displaystyle $$


 * }


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$$ = \underbrace{\int_{\Omega} \frac{ \partial }{\partial x_i} \left ( w \kappa_{ij} \frac{ \partial u }{\partial x_j} \right ) d \Omega}_{\color{Blue} divergence \ theorem} - \int_{\Omega} \frac{ \partial w}{\partial x_i} \left ( \kappa_{ij} \frac{ \partial u }{\partial x_j} \right ) d \Omega \ $$
 *  $$ \displaystyle $$


 * }


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$$ = \underbrace{ \int_{\Omega}}_{\color{Blue}\Gamma_g \cup \Gamma_h} \underbrace{ n_i \left ( w \kappa_{ij} \frac{ \partial u }{\partial x_j} \right ) }_{\color{Blue} w \underbrace{\mathbf{n} \cdot \boldsymbol{\kappa} \cdot grad(u) }_{\color{Blue} {\color{Red} -} \mathbf{n} \cdot \mathbf{q}}} d (\partial \Omega) - \int_{\Omega} \underbrace{ \frac{ \partial w}{\partial x_i} \kappa_{ij} \frac{ \partial u }{\partial x_j} }_{\color{Blue} \underbrace{(grad(w))}_{\nabla w} \cdot \boldsymbol{\kappa} \cdot \underbrace{ grad(u)}_{\nabla u} } d \Omega \ $$
 *  $$ \displaystyle $$


 * }

[[media: Fe1.s11.mtg34.djvu| Page 34-3 ]]


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$$ {\color{Blue}\mathbb{K}} = {\color{Red}-} \int_{\Gamma_g} w \ \mathbf{n} \cdot \mathbf{q} \ d \Gamma_g {\color{Red}-} \int_{\Gamma_h} w \underbrace{ \mathbf{n} \cdot \mathbf{q}}_{\color{Blue}h} \ d \Gamma_h + \int_{\Omega} \triangledown w \cdot \boldsymbol{\kappa} \cdot \triangledown u \ d \Omega \ $$
 *  $$ \displaystyle $$


 * }


 * }


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$$ {\color{Blue}\mathbb{K}} = {\color{Red}-} \int_{\Gamma_g} \cancel{w}^{=0} \ \mathbf{n} \cdot \mathbf{q} \ d \Gamma_g {\color{Red}-} \int_{\Gamma_h} w \underbrace{ \mathbf{n} \cdot \mathbf{q}}_{\color{Blue}h} \ d \Gamma_h + \int_{\Omega} \triangledown w \cdot \boldsymbol{\kappa} \cdot  \triangledown u \ d \Omega \ $$

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


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WF: [[media: Fe1.s11.mtg34.djvu| (1) page 34-2, (2)  ]] $$ \Rightarrow \ $$

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$$ \tilde{k}(w, u) = \int_{\Omega} \triangledown w \cdot \boldsymbol{\kappa} \cdot  \triangledown u \ d \Omega \ $$

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

<span id="(1)">
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$$ \tilde{f}(w) = - \int_{\Gamma_{\color{Red}h}} w \ h \ d \Gamma_{\color{Red}h} \ + \int_{\Omega} w \ g \ d \Omega $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(6)}
 * }


 * }

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(7)}
 * } $$ \ $$


 * }

[[media: Fe1.s11.mtg34.djvu| Page 34-4 ]]

Computation of $ {\color{Blue}\mathbf{k}} ^{\color{Red}e} \ $ in parent coordinates $ {\color{Blue} \left \{ \xi \right \} } $ : |undefined figure [[media: Fe1.s11.mtg30.djvu| page 30-1 ]]

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$$ d \Omega_{\color{Blue}x} := d{\color{Blue}x} _1 d{\color{Blue}x} _2 \ $$ infinitesimal area in $$ \left \{ {\color{Blue}x} _i \right \} \ $$  physical  coordinates

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

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$$ d \Omega_{ {\color{Blue}\xi} } := d {\color{Blue}\xi}_1 d {\color{Blue}\xi}_2 \ $$ infinitesimal area in $$ \left \{{\color{Blue}\xi}_i \right \} \ $$ parent coordinates

$$
 * <p style="text-align:right"> $$ \displaystyle
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$$ d \Omega_{\color{Blue}x} = det \mathbf{J}({\color{Blue}\xi}) \ d \Omega_{\color{Blue}\xi} \ $$

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

Jacobian Matrix
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$$ \mathbf{J}_{\color{Red}2x2} (\xi) = \left [ \frac{\partial x_{\color{Blue}i} }{\partial \xi_{\color{Red}j} } \right ] =

\begin{bmatrix} \frac{\partial x_{\color{Blue}1}}{\partial \xi_{\color{Red}1}} & \frac{\partial x_{\color{Blue}1}}{\partial \xi_{\color{Red}2}}    \\

& \\

\frac{\partial x_{\color{Blue}2}}{\partial \xi_{\color{Red}1}} & \frac{\partial x_{\color{Blue}2}}{\partial \xi_{\color{Red}2}} \end{bmatrix}

$$

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

, where subscripts $$ {\color{Blue}i}, {\color{Red}j} \ $$ represents $$ {\color{blue}row } \ $$ and $$ {\color{Red}column} \ $$, respectively.

Similar to [[media: Fe1.s11.mtg30.djvu| (4) page 30-3 ]]

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$$ \underbrace{ \mathbf{x}^{\color{Red}e} }_{({\color{Blue}x_1}^{\color{Red}e}, {\color{Blue}x_2}^{\color{Red}e})} = \varphi^{\color{Red}e}( \underbrace{\boldsymbol{\xi}}_{\color{Blue}(\xi_1, \xi_2)}) = \sum_{\color{Blue}I=1}^{\color{Blue}n_e} N_{\color{Blue}I} ( \underbrace{\boldsymbol{\xi}}_{\color{Blue}(\xi_1, \xi_2)}) \underbrace{ \mathbf{x}^{\color{Red}e}_{\color{Blue}I}}_{\color{Blue} (x_{1,I}^{\color{Red}e}, x^{\color{Red}e} _{2,I} ) } \ $$

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

, where $$ {\color{Blue}x_{1,I}}^{\color{Red}e} = {\color{Blue} x_{direction, \ element \ node \ number}^{\color{Red}e} }\ $$

[[media: Fe1.s11.mtg34.djvu| Page 34-5 ]]

For each element


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$$ N_{\color{Blue}I} (.) \ $$ similar to [[media: Fe1.s11.mtg29.djvu| (1) page 29-2 ]]