# 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  :

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

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

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

### Page 15-2

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

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

### Page 15-3

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

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

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

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

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

### Page 15-4

Euler Equations: Special Homogeneous Ln_ODE_VC

$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 $y'=y^{(1)}\$  and $y=y^{(0)}\$

Two methods of solution:

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

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

(Or Trial Solution) K.etal(2003)