User:Eml4500.f08.ateam.rivero/w6

Honesty, Imagination, Ethics
The following quotes are excerpts from Dr. L. Vu-Quoc main homework page for the Finite Element Analysis: EML 4500 course at the University of Floida.

Cont. PVW to Discrete PVW (Cont'd)
Langrangian Interpolation

Motivation for the form of $$ N_{i}(x) $$ and $$ N_{i+1}(x) $$

1. $$ N_{i}(x) $$ and $$ N_{i+1}(x) $$ are linear (straight line) thus any linear combo of $$ N_{i} $$ and $$ N_{i+1} $$ is also linear, and in particular expression for $$ u(x) $$ on p.31-3.

$$ N_{i}(x) = \alpha_{i} + \beta_{i}(x) $$, where $$ \alpha_{i} , \beta_{i} $$, are real numbers $$ N_{i+1}(x) = \alpha_{i+1} + \beta_{i+1}(x) $$, where $$ \alpha_{i+1} , \beta_{i+1} $$, are real numbers $$ N_{i}*d_{i}+N_{i+1}*d_{i+1} = (\alpha_{i}+\beta_{i}x)d_{i}+(\alpha_{i+1}+\beta_{i+1}x)d_{i+1} $$ ;is clearly a linear function in x 2. Recall equation for $$ u(x) $$ $$ u(xi) = N_{i}(xi)d_i+N_{i+1}(xi)d_{i+1} $$ Where N_{i}(xi)d_i = 1 and N_{i+1}(xi)d_{i+1} = 0