User:Eml5526.s11.team5.JA/Mtg2

Mtg 2: Fri, 7 Jan 11

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3) Spectral methods (EGM 6341): Highly accurate, simple geometries. For complex geometries - spectral element methods (ideas from FEM)

Mathematical models of engineering problems


where:

$$ \Gamma_u = \Gamma_g \ $$ (Hughes 2000) $$ \rightarrow $$ Essential boundary condition: prescribed unknown u(x,t)

$$ \Gamma_h \ $$ $$ \rightarrow $$ Natural boundary condition: prescribed flux or force

$$ \underbrace{\partial \Omega}_{\color{Blue}boundary \ of \ \Omega } = \Gamma_u \cup \Gamma_h \ $$

,where $$ \cup \ $$ denotes "union"

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$$ \mathbf{u}(x,t) = \ $$ vector field
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$$
 *  $$ \displaystyle
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2) Boundary conditions:

a) Essential boundary condition on $$ \Gamma_u \ $$

b) Natural boundary condition $$ \Gamma_h \ $$


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Example: Elastic bar



, $$ {\color{Blue}E(x)} \ $$  is Young's Modulus

Static:
1) PDE (actually ODE for 1D static)


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$$ \frac{d}{dx} \left [ E(x) A(x) \frac{du}{dx} \right ] + f(x,t) = 0 \ $$ (cf. F&B p. 44)

$$
 *  $$ \displaystyle
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2) Boundary Conditions

a) Essential (displacement) boundary condition:


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$$ u(0) = \underbrace{\bar{u}}_{\color{Red}constant \ and \ not necessary \ 0} {\color{Purple} = g } \ $$

$$
 *  $$ \displaystyle
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b) Natural (force) boundary condition:


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$$ N(L) = \sigma(L) \ A(L) = F \Rightarrow \frac{du(L)}{dx} = \frac{F}{E(L) \ A(L)} $$

$$
 *  $$ \displaystyle
 * }

, with

$$ \sigma(L) = E(L) \  \epsilon(L) \ $$ and   $$\epsilon(L) = \frac{du(L)}{dx}  \ $$

Also $$ \frac{F}{E(L) \ A(L)} \ $$ is defined as $$ {\color{Blue} h } \ $$ in Hughes 2000.

Dynamic:
1) PDE (actually ODE for 1D static)


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$$ \frac{d}{dx} \left [ E{\color{Blue}(x)} A{\color{Blue}(x)} \frac{du}{dx} \right ] {\color{Blue}+} f(x,t) = \underbrace{m(x)}_{\color{Blue} \rho(x) \ A(x)} \frac{\partial^2 u}{\partial t^2} \ $$

$$
 *  $$ \displaystyle
 * }

2) Boundary conditions

a) Essential (displacement) boundary condition:


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$$ u(x=0,t) = \bar{u}(t) \ $$

$$
 *  $$ \displaystyle
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b) Natural (force) boundary condition:


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$$ \frac{\partial u(x = L,t)}{\partial x} = \frac{F(t)}{E(L).A(L)} \ $$

$$
 *  $$ \displaystyle
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3) Initial conditions


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$$ u(x,t=0) = u_0(x) \ $$

$$
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<span id="(1)">
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$$ \frac{ \partial u(x ,t = 0)}{\partial x} = v_0(x) \ $$

$$
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End Example