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  :

File:EGM6321 F10 TEAM1 WILKS xc mtg15 1.svg

$N_J(x)=$ finite element basis function associated with node J = "hat" function.

File:EGM6321 F10 TEAM1 WILKS xc mtg15 2.svg

$N_J\in C^0$ , but $N_J\notin C^1$

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

Page 15-2

File:EGM6321 F10 TEAM1 WILKS xc mtg15 3.svg

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

File:EGM6321 F10 TEAM1 WILKS xc mtg15 4.svg

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

Page 15-3

File:EGM6321 F10 TEAM1 WILKS xc mtg15 5.svg

$L_2(y_H^{\alpha })=0$ , $\alpha =1,2$

$L_2(y_P)=f(x)$

$y=A{y}_H^1+B{y}_H^2+y_P$ , where A and B are constants

Where this can be rewritten for x as: $x=A{x}_H^1+B{x}_H^2+x_P$ , where A and B are constants

$L_2(y)=f\iff y=L_2(f)$

Page 15-4

Euler Equations: Special Homogeneous Ln_ODE_VC

$a_n{x}^n{y}^{(n)})+a_{n-1}{x}^{n-1}{y}^{n-1}+...+a_{1}x{y}^{\prime }+a_{0}y=0\iff \sum _{i=0}^n{a}_i{x}^i{y}^{(i)}=0$

Where $y^{\prime }=y^{(1)}$ 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)

References

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