User:EGM6321.F10.TEAM1.WILKS/Mtg15

=EGM6321 - Principles of Engineering Analysis 1, Fall 2009= Mtg 15: Thur, 24Sept09

[[media: Egm6321.f09.mtg15.djvu | Page 15-1]]
[[media: Egm6321.f09.mtg14.djvu | Ex.1 P.14-4 ]] : $$ N_J(x)= \ $$ finite element basis function associated with node J = "hat" function. $$ N_J \in C^0 \ $$, but $$ N_J \notin C^1 \ $$ [[media: Egm6321.f09.mtg14.djvu | Ex.1 P.14-4 ]]: Cubic Spline (Bexler, cubic Hermitian)

[[media: Egm6321.f09.mtg15.djvu | 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 $$ \alpha\ =1,2 \ $$ END HW

[[media: Egm6321.f09.mtg15.djvu | 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 $$ L_2(y)=f \Leftrightarrow y=L_2(f) \ $$

[[media: Egm6321.f09.mtg15.djvu | Page 15-4]]
Euler Equations: Special Homogeneous Ln_ODE_VC $$ a_nx^ny^{(n)})+a_{n-1}x^{n-1}y^{n-1}+...+a_1xy'+a_0y=0 \Leftrightarrow \sum_{i=0}^n a_ix^iy^{(i)}=0 \ $$ 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) [[media: http://books.google.com/books?id=9Cg3HWCnCjAC&printsec=frontcover&dq=differential+equations+billingham&ei=pGR4SpPVLojSMpb07Qw#v=onepage&q=&f=false | K.etal(2003) ]]