User:Eml5526.s11.team5.JA/Mtg38

Mtg 38: Wed, 30 Mar 11

[[media: Fe1.s11.mtg38.djvu| Page 38-1 ]]


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On $$ \Gamma_g: \ u(\mathbf{x}, t) = g(\mathbf{x}, t) \ \forall \ \mathbf{x} \in \Gamma_g \ $$

$$
 *  $$ \displaystyle {\color{Red}(2)}
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On $$ \Gamma_h: \ \mathbf{q}(\mathbf{x}, t) \cdot \mathbf{n}(\mathbf{x}) = h(\mathbf{x}, t) \ \forall \ \mathbf{x} \in \Gamma_h \ $$

$$
 *  $$ \displaystyle
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$$ = - \mathbf{n} (\mathbf{x}) \cdot \boldsymbol{\kappa}(\mathbf{x}) \cdot grad \ u (\mathbf{x}, t)$$

$$
 *  $$ \displaystyle {\color{Red}(3)}
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On $$ \Gamma_H: \ \mathbf{q}(\mathbf{x}, t) \cdot \mathbf{n}(\mathbf{x}) = H \left [ u( \mathbf{x}, t) - u_{\color{Red}\infty} \right ] \ \forall \ \mathbf{x} \in \Gamma_H \ $$

$$
 *  $$ \displaystyle
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$$ = - \mathbf{n} (\mathbf{x}) \cdot \boldsymbol{\kappa}(\mathbf{x}) \cdot grad \ u (\mathbf{x}, t)$$

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

Note:

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

Essential boundary condition [[media: Fe1.s11.mtg34.djvu| (2) page 34-1  ]]

Natural boundary condition [[media: Fe1.s11.mtg34.djvu| (3) page 34-1  ]]

[[media: Fe1.s11.mtg38.djvu| Page 38-2 ]]

1) Develop Weighted Form (WF)  similar to [[media: Fe1.s11.mtg34.djvu| (3)-(6) page 34-3  ]]

2) Develop semidiscrete equations (ODEs)  similar to [[media: Fe1.s11.mtg23.djvu| (3) page 23-3  ]]. Give detailed expressions for all quantities. (see [[media: Fe1.s11.mtg22.djvu|  Mtg 22  ]])

G2DM2.0/D1:


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$$ \frac{\partial u}{\partial t} = 0, g = 2, h = 3 \ $$

$$
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$$ H = 0.5, u_{\color{Red}\infty} = 0.1 \ $$

$$
 *  $$ \displaystyle
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 * } $$ \ $$
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3) Solve G2DM2.0/D1  using 2D LIBF (similar to  HW 6.6 ) till $$ 10^{-6} \ $$ accuracy at center. Plot solution in 3D with contours.
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Note: Two conventions for Jacobian matrix, [[media: Fe1.s11.mtg34.djvu|  page 34-4  ]]

[[media: Fe1.s11.mtg34.djvu| (2) page 34-4 ]]:


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

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

, where $$ {\color{Blue}i}, {\color{Red}j} \ $$ represent $$ {\color{Blue}row}, {\color{Red}column} \ $$, respectively

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<span id="(1)">
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$$ \mathbf{J}_{\color{Red}FB} = \left [ \frac{ \partial x_{\color{Red}j}}{\partial \xi_{\color{Blue}i}} \right ] \ $$, FB, page 167, (7.32)

$$
 * <p style="text-align:right"> $$ \displaystyle
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$$ \mathbf{k}^{\color{Red}e} \ $$ is the same for any convention since

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$$ det \ \mathbf{J} = det \ \mathbf{J}^{\color{Red}T}_{\color{Red}FB} = det \ \mathbf{J}_{\color{Red}FB} \ $$

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

[[media: Fe1.s11.mtg35.djvu| (3) page 35-3: ]]

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$$ \triangledown_{\color{Blue}x} = \mathbf{J}^{\color{Red}-T}( \boldsymbol{\xi} ) \triangledown_{\color{Blue}\xi} = \underbrace{ \mathbf{J}^{\color{Red}-1}_{\color{Red}FB} ( \boldsymbol{\xi} ) \triangledown_{\color{Blue}\xi} }_{\color{Blue}FB, page \ 167, (7.32) - (5.34)} \ $$

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

Reference: http://en.wikiversity.org/wiki/User:Egm6936.f09/Gradient_of_vector:_Two_tensor_conventions

Note:

p-convergence: Increase degree of interpolating polynomial, example HW 5.1

h-convergence: Increase number of elements, example HW 5.7

Comments on HW 5.7 Team 5

[[media: Fe1.s11.mtg38.djvu| Page 38-4 ]]

Membrane:



Static: