User:Eml5526.s11.team5.JA/Mtg4

Mtg 4: Wed, 12 Jan 11

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Other self - adjoin problems:

Heat problems (diffusion equation as particular case)

F&B p.46, p.59

steady - state (static):

1-D (F & B p.46)

PDE: $$

\frac{d}{dx} \left ( A(x)k(x) \frac{d{\color{Blue}u}}{dx} \right ) + {\color{Blue}f} = 0 $$


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$$ u(x) = T(x) {\color{Blue} \leftarrow Temperature} \ $$

$$
 *  $$ \displaystyle
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$$ f(x) = s(x) {\color{Blue} \leftarrow distributed \ heat \ source } \ $$

$$
 *  $$ \displaystyle
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$$ k(x) = {\color{Blue} conductivity} \ x \in ]0,L[ = \Omega \ $$

$$
 *  $$ \displaystyle
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Boundary conditions:


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$$ -q(0) = k \frac{du(0)}{dx} = \bar{q} \ at \ x=0 \ $$

$$
 *  $$ \displaystyle
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$$ u(L) = T(L) = \bar{q} $$

$$
 *  $$ \displaystyle
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Transient (dynamic):

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PDE: $$

\frac{d}{dx} \left ( A(x) \ k(x) \ \frac{du}{dx} \right ) + f(x,t) = \underbrace{ A \ \rho}_{\color{Blue} m = A \ \rho}  c \underbrace{\frac{\partial u}{\partial t}}_{\color{Blue} 1st \ order} \ $$

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

2-D and 3-D:

Transient:


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PDE: $$

div \left [ k(x) grad(u) \right ] + f(x,t) = \rho c \frac{\partial u}{\partial t} \ $$

$$
 *  $$ \displaystyle {\color{Red}(2)}
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$$ \Omega \in R^{nd} \ $$

$$
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$$

nd = \begin{cases} 2, & {\color{Blue} 2-D } \\ 3, & {\color{Blue} 3-D } \end{cases}

$$

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

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$$ x \in \Omega \ $$

$$
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$$ x = (x_1, x_2, x_3) \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle
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Note: I-D PDE = After integration over cross section, example, PDE is valid at any point inside cross section (example, at point 'x')

2-D and 3-D PDE valid $$ x \in R^{nd} \ $$

elastodynamics (see F&B p. 223 for static case):

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$$ div \boldsymbol{\sigma} + \mathbf{b} = \rho \frac{\partial^2 \mathbf{u}}{\partial t^2} \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(3)}
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$$ \boldsymbol{\sigma}(x,t) = \mathbf{D}(x) \boldsymbol{\varepsilon}(x,t) $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(4)}
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$$ \boldsymbol{\varepsilon} = \frac{1}{2} \left ( grad(\mathbf{u}) + grad^T(\mathbf{u}) \right ) \ $$

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

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$$ div \boldsymbol{\sigma} = \frac{1}{2} div \left [\mathbf{D} \left ( grad(\mathbf{u}) + grad^T(\mathbf{u}) \right ) \right ] $$

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

"conferre" (cf.) (6) to first term of [[media: Fe1.s11.mtg4.djvu| (1) - (2)  page 4-2 ]] (Montesquieu)

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Generic (abstract, general) equation for 1-D elastic bar, 1-D heat (diffusion) F&B see. 3.1.3, p.46

"Strong" form:

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$$

{\color{Red}s} = \begin{cases} {\color{Red}s} = 2, & for \ elastic \ bar\\ {\color{Red}s} = 1, & for \ heat \end{cases}

$$

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

Boundary conditions

on $$ \Gamma_g : \ \frac{u}{\Gamma_g} = g \leftarrow {\color{Blue}essential} \ boundary \ condition \ $$

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$$ \Gamma_g \ $$ = boundary where essential boundary condition is applied

$$
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on $$ \Gamma_h: \  \underbrace{ \left ( n. a_2 \frac{\partial u}{\partial x} \right ) }_{\color{Red} 'flux'} |_{\Gamma_h} = h \leftarrow  {\color{Blue}natural } \ boundary \ condition  \ $$

Initial conditions


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