University of Florida/Egm6341/s10.team3.aks/HW6

(3) Evaluate the rest of the coefficient of matrix
Ref Lecture Notes [[media:Egm6341.s10.mtg32.djvu|p.35-3]]

Problem Statement
Evaluate the remaining coefficient of Matrix by using degrees of Freedom

Solution
We have

$$ Z(s)=\sum_{i=0}^{3}c_is^i=\sum_{i=0}^{3}N_i(s)d_i $$

such that

$$ d_1=Z_i$$

$$ d_2=\dot{Z_i}$$

$$ d_3=Z_{i+1}$$

$$ d_4={\dot{Z}_{i+1}}$$

where

$$ \dot{Z}=\frac{dZ}{dt}={\frac{dZ}{ds}} \frac{ds}{dt} $$

such that $$ \frac{ds}{dt} = \frac {1}{h} $$

We know the coefficient of matrix for first two rows from lecture notes   [[media:Egm6341.s10.mtg32.djvu|p.35-3]]

$$\begin{bmatrix} 1 &0 &0  &0 \\  0&1  &0  &0 \end{bmatrix}$$

Using the equations above we have

$$  d_3=Z_{i+1}=Z(s=1)=c_0+c_1+c_2+c_3 $$

$$  d_4=\dot{Z}_{i+1}=\frac{1}{h}Z'(s=1)=c_1+2c_2+3c_3 $$

Putting the results in matrix form we obtain

$$\begin{bmatrix} 1 &0 &0  &0 \\ 0 &1  &0  &0 \\ 1 &1  &1  &1 \\ 0 &1  &2  &3 \end{bmatrix}\begin{Bmatrix} c_0\\ c_1\\ c_2\\

c_3\end{Bmatrix}= \begin{Bmatrix} Z_i\\ Z'_i\\ Z_{i+1}\\ Z'_{i+1} \end{Bmatrix}$$

(4) Verify the inverse of matrix using Matlab
Ref Lecture Notes [[media:Egm6341.s10.mtg32.djvu|p.35-4]]

Problem Statement
Find the inverse of given Matrix

A =

1    0     0     0     0     1     0     0     1     1     1     1     0     1     2     3

Solution
which is same as the one given on [[media:Egm6341.s10.mtg32.djvu|p.35-4]]

Hence Verified

(5) Identify basis functions and plot them
Ref Lecture Notes [[media:Egm6341.s10.mtg32.djvu|p.35-4]]

Problem Statement
Identify the basis functions

$$ N_i (s) $$

where $$ i = 1,2,3,4$$

Solution
We have

$$ z(s) = \sum_{i=0}^3 c^i s^i = \sum_{i=1}^4 N_i(s) d_i $$

Expanding above we obtain

$$ N_1 d_1 + N_2 d_2 + N_3 d_3 + N_4 d_4 = C_0 s^0 + C_1 s^1 + C_2 s^2 + C_3 s^3$$

$$                                      = C_0  + C_1 s^1 + C_2 s^2 + C_3 s^3$$

$$d_1 = z_i = z(s=0) = C_0$$

$$d_2 = \dot z_i = \dot z_i (s=0) = C_1$$

$$d_3 = z_{i+1}= z_{i+1} (s=1) = C_0 + C_1 + C_2  + C_3 $$

$$ d_4 = \dot z_{i+1} =C_1 + 2 C_2 + 3 C_3 $$

Inserting above values in first eq we obtain

$$(N_1 + N_3) C_0 + (N_2+N_3+N_4) C_1 + (N_3+2 N_4)C_2 + (N_3 + 3 N_4) C_3 = C_0 + C_1 s + C_2 s^2 + C_3 s^3$$

Comparing both LHS and RHS we obtain

$$N_1 + N_3 = 1$$

$$N_2+N_3+N_4 = s$$

$$N_3+2 N_4=s^2$$

$$N_3 + 3 N_4 = s^3$$

Solving above we obtain basis functions

$$N_1 = 1- 3 s^2 +2 s^3$$

$$N_2 = s^3 - 2 s^2 + s $$

$$N_3 = 3 s^2 - 2 s^3$$

Below is the plot of above basis functions



$$ N_4 = s^3 - s^2$$

= (6) Show that s = s(t) =

Ref Lecture Notes [[media:Egm6341.s10.mtg32.djvu|p.36-1]]

Problem Statement
We have to show that s is the function of t (s = s(t) )

Solution
We have (from [[media:Egm6341.s10.mtg32.djvu|p.35-1]] eq (1))

$$ t(s)= (1-s) t_i + s t_{i+1} $$

$$    = t_i + s (t_{i+1} - t_i ) $$

$$ but \frac {ds}{dt} = \frac{1}{h} $$

$$ or \frac {dt}{ds} = h $$

so $$ t_{i+1} - t_i = \frac{t_{i+1} - t_i}{1-0}= h $$

$$ t(s) = t_i +h s $$ $$ h s = t - t_i $$ $$ s = \frac {(t - t_i)} {h} $$ $$ s = s(t) $$

Hence Proved