User:Eml5526.s11.team01.roark/Mtg41

[[media: Fe1.s11.mtg41.djvu| Mtg 41]]: Wed, 13 Apr 11 [[media: Fe1.s11.mtg41.djvu| Page 41-1]] HW 7.2:

Continued

2) Transient Analysis:

Initial Condition

$$\displaystyle u\left( \mathbf x,t=0 \right)=u_{s}^{h}\left(\mathbf x \right),~\forall \mathbf x\in \text{ }\!\!\Omega\!\!\text{ }$$

$$\displaystyle \dot{u}\left(\mathbf x,t=0 \right)=0,~\forall \mathbf x\in \text{ }\!\!\Omega\!\!\text{ }$$

$$\displaystyle T=4,~f\left(\mathbf x,t \right)=0$$in $$\displaystyle \text{ }\!\!\Omega\!\!\text{ }$$

Plot $$\displaystyle {{u}^{h}}\left( \underbrace{\mathbf 0}_{Center},t \right)$$ vs t for $$\displaystyle t\in \left[ 0,2{{T}_{1}} \right]$$

Produce a movie of the vibrating membrane.

End HW 7.2

Semidiscrete equation

Similar to [[media: Fe1.s11.mtg23.djvu|(3) p. 23-3]] with s=2

$$\displaystyle \mathbf M\ddot{\mathbf d}+\mathbf K \mathbf d=\mathbf F$$

$$\displaystyle \mathbf z:=\left[ \begin{matrix} \mathbf d \\ {\dot{\mathbf d}} \\ \end{matrix} \right]=\left[ \begin{matrix} {{\mathbf z}_{1}} \\ {{\mathbf z}_{2}} \\ \end{matrix} \right]\Rightarrow \dot{\mathbf z}=\left[ \begin{matrix} {\dot{\mathbf d}} \\ {\ddot{\mathbf d}} \\ \end{matrix} \right]=\left[ \begin{matrix} {{\mathbf z}_{2}} \\ {{\mathbf M}^{-1}}\left(\mathbf F-\mathbf K\underbrace{\mathbf d}_ \right) \\ \end{matrix} \right]$$

$$\displaystyle \mathbf z\left( 0 \right)=\left[ \begin{matrix} \mathbf d\left( 0 \right) \\ \dot{\mathbf d}\left( 0 \right) \\ \end{matrix} \right]$$

Matlab ode45 [[media: Fe1.s11.mtg41.djvu| Page 41-2]]

HW 7.2:

continued

3) Transient Analysis: Similar to (2) but with non-symmetric u_0

$$\displaystyle u\left(\mathbf x,t=0 \right)=\left( x+1 \right)\left( y+\frac{1}{2} \right)cos\left( \frac{\pi }{2}x \right)cos\left( \frac{\pi }{2}y \right),~\forall \mathbf x\in \text{ }\!\!\Omega\!\!\text{ }$$

Non-symmetric “bubble” (should produce sloshing vibration)





plot {(x+1)*(y+0.5)*cos(x*pi/2)*cos(y*pi/2)},-1<=x<=1,-1<=y<=1 [[media: Fe1.s11.mtg41.djvu| Page 41-3]]

$$\displaystyle u\left( \mathbf x,t=0 \right)=0,~\forall \mathbf x\in \text{ }\!\!\Omega\!\!\text{ }$$

$$\displaystyle T=4,~f\left(\mathbf x,t \right)=0$$in $$\displaystyle \text{ }\!\!\Omega\!\!\text{ }$$

Plot $$\displaystyle {{u}^{h}}\left( \underbrace{\mathbf 0}_{Center},t \right)$$ vs t for $$\displaystyle t\in \left[ 0,2{{T}_{1}} \right]$$

Produce a movie of the vibrating membrane.

End HW 7.2

Triangular element as quad element with 2 identical nodes



Local node list = {1,2,3,4}

Global node list={205, 51, 51,102}

[[media: Fe1.s11.mtg41.djvu| Page 41-4]]

HW 7.3:

Let $$\displaystyle \text{ }\!\!\Omega\!\!\text{ }$$=circle with unit radius. Find static solution such that T=4, $$\displaystyle f\left(\mathbf x \right)=1$$ in $$\displaystyle \text{ }\!\!\Omega\!\!\text{ }$$, g=0 on $$\displaystyle {{\Gamma }_{g}}=\partial \Omega $$ to 〖10〗^(-6) accuracy @ center. Use both quad and triangular element. Symmetric => Use ¼ of meshes. Plot deformed shape in 3-D.

End HW 7.3

[[media: Fe1.s11.mtg41.djvu| Page 41-5]]

Comments on HW6 continued:

HW6.10, mixed boundary conditions.

HW 7.4:

Redo HW 6.10.

End HW 7.4


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WF: Similar to [[media: Fe1.s11.mtg34.djvu|(7) p. 34-3]], with
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$$\displaystyle \tilde{k}\left( w,u \right)=\underset{\text{ }\Omega \text{ }}{\mathop{\overset – {\mathop{\int }}\,}}\,\nabla \text{w}\cdot \mathbf{K}\cdot \nabla \text{ud}\Omega \text{ +}\underset{\text{ }\Gamma {{}_{\text{H}}}}{\mathop{\overset – {\mathop{\int }}\,}}\,\text{wHud}{{\Gamma }_{\text{H}}}$$
 *  (1)
 * }
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 * {| style="width:100%" border="0"

$$\displaystyle \tilde{f}\left( w \right)=\underset{\overset – {\mathop -\int }}\,\text{whd}{{\text{ }\!\!\Gamma\!\!\text{ }}_{\text{h}}}+\underset{\overset – {\mathop \int }}\,\text{wH}{{\text{u}}_{\infty }}\text{d}{{\text{ }\!\!\Gamma\!\!\text{ }}_{\text{H}}}\underset{\text{ }\!\!\Omega\!\!\text{ }}{\overset – {\mathop \int }}\,\text{wfd }\!\!\Omega\!\!\text{ }$$
 *  (2)
 * }
 *  (3)
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$$\displaystyle {{\eta }_{F}}=\underbrace_{Nat\,bc}\cup \underbrace_{mixed\,bc}\cup \underbrace_{remaining}$$

[[media: Fe1.s11.mtg39.djvu|Ex 1 p. 39-4]]: $$\displaystyle {{\eta }_{R}}=\left\{ 6,7,10,11 \right\}$$

$$\displaystyle \tilde{\mathbf c}=\left[ \begin{matrix} \underline \\ {{\mathbf c}_{F}} \\ \end{matrix} \right]=\left[ \begin{matrix} \underline \\ \begin{matrix} {{\mathbf c}_{h}} \\ {{\mathbf c}_{H}} \\ {{\mathbf c}_{R}} \\ \end{matrix} \\ \end{matrix} \right]$$ and $$\displaystyle \tilde{\mathbf d}=\left[ \begin{matrix} \underline \\ {{\mathbf d}_{F}} \\ \end{matrix} \right]=\left[ \begin{matrix} \underline \\ \begin{matrix} {{\mathbf d}_{h}} \\ {{\mathbf d}_{H}} \\ {{\mathbf d}_{R}} \\ \end{matrix} \\ \end{matrix} \right]$$

Where $$\displaystyle {{\mathbf d}_{h}}$$ and $$\displaystyle {{\mathbf d}_{H}}$$ overlap$$\displaystyle {{d}_{13}}$$, and $$\displaystyle {{\mathbf c}_{E}}$$ and $$\displaystyle {{\mathbf c}_{H}}$$ overlap$$\displaystyle {{c}_{13}}$$. [[media: Fe1.s11.mtg41.djvu| Page 41-6]]



$$\displaystyle {{\mathbf c}_{E}}=\left[ \begin{matrix} {{c}_{1}} \\ {{c}_{2}} \\ {{c}_{3}} \\ \begin{matrix} {{c}_{4}} \\ {{c}_{8}} \\ {{c}_{12}} \\ {{c}_{16}} \\ \end{matrix} \\ \end{matrix} \right]$$

$$\displaystyle {{\mathbf c}_{h=}}\left[ \begin{matrix} {{c}_{14}} \\ {{c}_{15}} \\ {{c}_{13}} \\ \end{matrix} \right]$$

Matrices overlap with $$\displaystyle {{c}_{13}}$$

$$\displaystyle {{\mathbf c}_{H=}}\left[ \begin{matrix} {{c}_{13}} \\ {{c}_{5}} \\ {{c}_{9}} \\ \end{matrix} \right]$$

$$\displaystyle {{\mathbf c}_{R}}=\left[ \begin{matrix} {{c}_{6}} \\ {{c}_{7}} \\ {{c}_{10}} \\ {{c}_{11}} \\ \end{matrix} \right]$$ [[media: Fe1.s11.mtg41.djvu| Page 41-7]]

From [[media: Fe1.s11.mtg41.djvu|(1) p. 41-4]]

$$\displaystyle \underset{\text{ }\Omega \text{ }}{\mathop{\overset – {\mathop{\int }}\,}}\,\nabla {{w}^{h}}\cdot \mathbf K\cdot \nabla {{u}^{h}}d\text{ }\Omega \text{ }={{\left[ \begin{matrix} \underline \\ {{\mathbf c}_{h}} \\ {{\mathbf c}_{H}} \\ {{\mathbf c}_{R}} \\ \end{matrix} \right]}^{T}}\left[ \begin{matrix} {{\mathbf K}_{EE}} & {{\mathbf K}_{Eh}} & {{\mathbf K}_{EH}} & {{\mathbf K}_{ER}} \\ {{\mathbf K}_{hE}} & {{\mathbf K}_{hh}} & {{\mathbf K}_{hH}} & {{\mathbf K}_{hR}} \\ {{\mathbf K}_{HE}} & {{\mathbf K}_{Hh}} & \mathbf K_{HH}^{H} & {{\mathbf K}_{HR}} \\ {{\mathbf K}_{RE}} & {{\mathbf K}_{Rh}} & {{\mathbf K}_{RH}} & {{\mathbf K}_{RR}} \\ \end{matrix} \right]\left[ \begin{matrix} \underline \\ {{\mathbf d}_{h}} \\ {{\mathbf d}_{H}} \\ {{\mathbf d}_{R}} \\ \end{matrix} \right]$$

From [[media: Fe1.s11.mtg41.djvu|(1) p. 41-4]]

$$\displaystyle \underset{\overset – {\mathop \int }}\,wHud{{\text{ }\!\!\Gamma\!\!\text{ }}_{H}}={{\left[ \begin{matrix} \underline{{{\mathbf c}_{E}}} \\ {{\mathbf c}_{h}} \\ {{\mathbf c}_{H}} \\ {{\mathbf c}_{R}} \\ \end{matrix} \right]}^{T}}\left[ \begin{matrix} {} & {} & {} & {} \\   {} & {} & {} & {}  \\   {} & {} & \mathbf K_{HH}^{H} & {}  \\ {} & {} & {} & {} \\ \end{matrix} \right]\left[ \begin{matrix} \underline \\ {{\mathbf d}_{h}} \\ {{\mathbf d}_{H}} \\ {{\mathbf d}_{R}} \\ \end{matrix} \right]$$

Taking a look at$$\displaystyle \mathbf K_{HH}^{H}$$, a 3x3 matrix, the top H is a conductance contribution from $$\displaystyle {{\text{ }\!\!\Gamma\!\!\text{ }}_{H}}$$, the first lower H is the rows $$\displaystyle {{\eta }_{H}}$$ and the second H is the columns $$\displaystyle {{\eta }_{H}}$$. [[media: Fe1.s11.mtg41.djvu| Page 41-8]] From [[media: Fe1.s11.mtg41.djvu|(2) p. 41-4]]:

$$\displaystyle \underset{\text{ }\!\!\Omega\!\!\text{ }}{\overset – {\mathop \int }}\,wfd\text{ }\!\!\Omega\!\!\text{ }={{\left[ \begin{matrix} \underline{{{\mathbf{c}}_{E}}} \\ {{\mathbf{c}}_{h}} \\ {{\mathbf{c}}_{H}} \\ {{\mathbf{c}}_{R}} \\ \end{matrix} \right]}^{T}}\left[ \begin{matrix} F_{E}^{f} \\ F_{h}^{f} \\ F_{H}^{f} \\ F_{R}^{f} \\ \end{matrix} \right]$$

From [[media: Fe1.s11.mtg41.djvu|(2) p. 41-4]]:

$$\displaystyle -\underset{\overset – {\mathop \int }}\,whd{{\text{ }\!\!\Gamma\!\!\text{ }}_{h}}={{\left[ \begin{matrix} \underline{{{\mathbf{c}}_{E}}} \\ {{\mathbf{c}}_{h}} \\ {{\mathbf{c}}_{H}} \\ {{\mathbf{c}}_{R}} \\ \end{matrix} \right]}^{T}}\left[ \begin{matrix} \underline –  \\ F_{h}^{h} \\ {} \\   {}  \\ \end{matrix} \right]$$

From [[media: Fe1.s11.mtg41.djvu|(2) p. 41-4]]:

$$\displaystyle \underset{\overset – {\mathop \int }}\,wH{{u}_{\infty }}d{{\text{ }\!\!\Gamma\!\!\text{ }}_{H}}={{\left[ \begin{matrix} \underline{{{\mathbf{c}}_{E}}} \\ {{\mathbf{c}}_{h}} \\ {{\mathbf{c}}_{H}} \\ {{\mathbf{c}}_{R}} \\ \end{matrix} \right]}^{T}}\left[ \begin{matrix} \underline –  \\ {} \\   F_{H}^{H}  \\ {} \\ \end{matrix} \right]$$

$$\displaystyle {{F}_{13}}=\begin{matrix} F_{H,13}^{f} \\ \equiv  \\ F_{h,13}^{f} \\ \end{matrix}+F_{h,13}^{h}+F_{H,13}^{H}$$