User:EGM6341.s11.team1.Chiu/S10 Mtg40

Mtg 40: Thu, 8 Apr 10

[[media: Egm6341.s10.mtg40.djvu| page40-1]]


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HW: Chaos, reproduce Figure 15.6 + 15.7 in King et. al. 2003 page 455-456


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Hints for solving logistic equation [[media: Egm6341.s10.mtg39.djvu| HW page39-1]]


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$$ \dot{x}=x(1-x) \ $$


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Separation of variables:  $$ \int^{x}_{x_{0}} \underbrace{ \frac{dx}{x(1-x)}}_{=x(t_{0})}= \int^{t}_{t_{0}} dt $$


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Partial fraction:  $$ \frac{1}{x(1-x)}= \frac{a}{x}+ \frac{b}{1-x} \ $$


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$$ a,b \ $$ are constant to be determined.


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End Hints for solving logistic equation

Element stability  FE + BE algorithms [[media: Egm6341.s10.mtg39.djvu| page39-2]]

FE: $$ \bar{x}_{i+1}= \underbrace{(1+hr)}_{ \color{blue} \rho_{F}} \bar{x}_{i}= \rho_{ \color{blue} F}( \rho_{ \color{blue} F} \bar{x}_{i-1})= \rho_{ \color{blue} F}^{i+1} \bar{x}_{0} \ $$


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Fix  $$ i, \ $$ consider   $$ h \to \infty \ $$


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$$ \rho_{ \color{blue} F}^{i+1} \to \infty \Rightarrow \bar{x}_{i+1} \to \infty \ $$  unstable


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BE: $$ \bar{x}_{i+1}= \underbrace{ \frac{1}{1-hr}}_{ \color{blue} \rho_{B}} \bar{x}_{i}= \rho_{B}^{i+1} \bar{x}_{0} \ $$

[[media: Egm6341.s10.mtg40.djvu| page40-2]]


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Fix  $$ i, \ $$ let   $$ h \to \infty \Rightarrow \left | \rho_{B}^{i+1} \right | \to \infty { \color{red} \to 0} \Rightarrow \left | \bar{x}^{i+1} \right | \to 0 \ $$ stable: numerical solution does not blow up $$ \color{blue} \forall h \ $$.


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End Element stability  FE + BE algorithms


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HW: Integrate Logistic equation ([[media: Egm6341.s10.mtg39.djvu|HW page39-1]])

Already found  $$ \hat{h} \ $$   at HS algorithm yields error $$ \theta(10^{-6}) \ $$

Run HS with $$ h=2^{k} \hat{h}. \ $$

Develop + run FE  $$ \ h=2^{k} \hat{h}. \ $$

Develop + run BE  $$ \ h=2^{k} \hat{h}. \ $$


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Equations of motion of aircraft: [[media: Egm6341.s10.mtg33.djvu| page33-3]]


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$$ \dot{ \mathbf{z}}= \mathbf{f}( \mathbf{z}, \mathbf{u}, t) \ $$  [[media: Egm6341.s10.mtg33.djvu| (3) page33-3]]


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$$ \int^{t_{i+1}}_{t_{i}} \dot{ \mathbf{z}}dt= \int^{t_{i+1}}_{t_{i}} \mathbf{f}( \mathbf{z}, \mathbf{u}, t)dt \ $$


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$$ \underset{ \color{red} 4 \times 1}{ \mathbf{z}_{i+1}}- \underset{ \color{red} 4 \times 1}{ \mathbf{z}_{i}}= \frac{ \frac{h}{2}}{3} \left [ \mathbf{f}_{i}+4 \mathbf{f}_{i+ \frac{1}{2}}+ \mathbf{f}_{i+1} \right ] \ $$


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$$ \Leftrightarrow \underset{ \color{red} 4 \times 1}{ \mathbf{F}} ( \mathbf{Z}_{i+1}) = \underset{ \color{red} 4 \times 1}{ \mathbf{0}} \ $$


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[[media: Egm6341.s10.mtg40.djvu| page40-3]]

NR algorithm [[media: Egm6341.s10.mtg38.djvu| page38-1]]:


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$$ \underset{ \color{red} 4 \times 1}{ \mathbf{z}_{i+1}^{(k+1)}}= \underset{ \color{red} 4 \times 1}{ \mathbf{z}_{i+1}^{(k)}}- \left [ \frac{d \mathbf{F}( \mathbf{z}_{i+1}^{(k)})}{d \mathbf{Z}} \right ]^{ \color{blue} -1}_{ \color{red} 4 \times 4} \underset{ \color{red} 4 \times 1}{ \mathbf{F}}( \mathbf{z}_{i+1}^{(k)}) \ $$


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$$ \mathbf{z}_{i+1}^{(0)}= \mathbf{z}_{i} \ $$


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End Equations of motion of aircraft

Review:  $$ \underset{ \color{red} m \times 1}{ \mathbf{f}( \mathbf{x})}= \left \lfloor f_{1}( \mathbf{x}),..., f_{m}( \mathbf{x}) \right \rfloor^{ \color{red} T} \ $$


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$$ \underset{ \color{red} n \times 1}{ \mathbf{x}}= \left \lfloor x_{1},..., x_{n} \right \rfloor^{ \color{red} T} \ $$


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$$ \underset{ \color{red} m \times n}{ \frac{d \mathbf{f}( \hat{ \mathbf{x}})}{d \mathbf{x}}}= \left [ \frac{df_{i}(\hat{ \mathbf{x}})}{dx_{j}} \right ] \ $$


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$$ i: \ $$  row index   $$ i=1,...,m, \ $$   $$ j: \ $$   column index   $$ j=1,...,n \ $$


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End Review


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HW:

S+Z 2007 parameters for $$ EQM \color{red} \to 0 \ $$

Table 1 page 315 S+Z 2007


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