University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg15

EGM6321 - Principles of Engineering Analysis 1, Fall 2009

Mtg 15: Thur, 24Sept09

Page 15-1

Ex.1 P.14-4  :

${\displaystyle N_{J}(x)=\ }$  finite element basis function associated with node J = "hat" function.

${\displaystyle N_{J}\in C^{0}\ }$  , but ${\displaystyle N_{J}\notin C^{1}\ }$

Ex.1 P.14-4 : Cubic Spline (Bexler, cubic Hermitian)

Page 15-2

Cubic ${\displaystyle \Rightarrow \ \ }$  4 Coefficients ${\displaystyle \Rightarrow \ \ }$  4 degrees of freedom per element ${\displaystyle \Rightarrow \ \ }$  2 degrees of freedom per node (each element has 2 nodes)

HW: ${\displaystyle N_{J}^{\alpha \ }(x)\in C^{1}\ }$ , but ${\displaystyle N_{J}^{\alpha \ }\notin C^{2}\ }$ , for ${\displaystyle \alpha \ =1,2\ }$

Page 15-3

${\displaystyle L_{2}(y_{H}^{\alpha \ })=0\ }$  , ${\displaystyle \alpha \ =1,2\ }$

${\displaystyle L_{2}(y_{P})=f(x)\ }$

${\displaystyle y=Ay_{H}^{1}+By_{H}^{2}+y_{P}\ }$  , where A and B are constants

Where this can be rewritten for x as: ${\displaystyle x=Ax_{H}^{1}+Bx_{H}^{2}+x_{P}\ }$  , where A and B are constants

${\displaystyle L_{2}(y)=f\Leftrightarrow y=L_{2}(f)\ }$

Page 15-4

Euler Equations: Special Homogeneous Ln_ODE_VC

${\displaystyle a_{n}x^{n}y^{(n)})+a_{n-1}x^{n-1}y^{n-1}+...+a_{1}xy'+a_{0}y=0\Leftrightarrow \sum _{i=0}^{n}a_{i}x^{i}y^{(i)}=0\ }$

Where ${\displaystyle y'=y^{(1)}\ }$  and ${\displaystyle y=y^{(0)}\ }$

Two methods of solution:

Method 1: Transfer of variables ${\displaystyle x=e^{t}\ }$

Method 2: Method of undetermined coefficients ${\displaystyle y=x^{r}\ }$

(Or Trial Solution) K.etal(2003)